#72343612

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#442
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₂²×C₈	C₂⁵
socle	C₂²	C₂²×#442
prime radical	C₂³×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ C₂²×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

4929 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 68 points:

(5,6,9,16,27,47,37,18,12,15,33,46,10,17,11,35)(7,19,22,39,34,51,59,41,21,48,58,13,23,40,24,50)(8,25,29,52,65,14,36,43,28,62,32,44,30,45,31,49)(20,54,56,42,64,53,61,38,55,67,60,68,26,63,57,66);(1,2)(3,4)(5,7)(6,13)(8,26)(9,24)(10,34)(11,22)(12,21)(14,42)(15,39)(16,48)(17,50)(18,51)(19,46)(20,30)(23,27)(25,38)(28,64)(29,60)(31,57)(32,56)(33,59)(35,40)(36,61)(37,58)(41,47)(43,54)(44,63)(45,68)(49,67)(52,53)(55,65)(62,66);(2,4)(5,8,27,65,12,28,10,30)(6,14,17,25,15,45,47,62)(7,20,34,64,21,55,23,26)(9,32,37,31,33,29,11,36)(13,38,41,42,39,66,50,68)(16,49,35,44,46,43,18,52)(19,53,40,54,48,63,51,67)(22,60,59,57,58,56,24,61);(7,21)(13,39)(19,48)(20,55)(22,58)(23,34)(24,59)(26,64)(38,66)(40,51)(41,50)(42,68)(53,63)(54,67)(56,60)(57,61);(6,15)(8,28)(13,39)(16,46)(17,47)(18,35)(19,48)(20,55)(26,64)(29,32)(30,65)(31,36)(40,51)(41,50)(56,60)(57,61)

Polycyclic presentation:

2^10|a^2=f,c^2=fhij,f^2=ij,g^2=j,h^2=i,i^2=j,b^a=bh,c^a=ci,c^b=cg,d^b=dj,e^a=ej,e^c=ej,f^b=fi,f^c=fj,g^b=gj,h^a=hj,h^b=hij,h^c=hj,i^b=ij

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343616, #72343873, #72343904
maximal normal subgroup of	#72343616, #72343873, #72343904

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.6);;
gap> SetPower(coll, 3, F.6*F.8*F.9*F.10);;
gap> SetPower(coll, 6, F.9*F.10);;
gap> SetPower(coll, 7, F.10);;
gap> SetPower(coll, 8, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.8);;
gap> SetConjugate(coll, 3, 1, F.3*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.9*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);