#72343632

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:

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

Polycyclic presentation:

2^10|a^2=f,b^2=g,c^2=fhi,f^2=ij,h^2=i,i^2=j,b^a=bh,c^a=ci,c^b=cgi,d^b=dj,d^c=dj,e^a=ej,e^b=ej,f^b=fi,f^c=fj,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	#72343634, #72343873, #72343897
maximal normal subgroup of	#72343634, #72343873, #72343897

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, 2, F.7);;
gap> SetPower(coll, 3, F.6*F.8*F.9);;
gap> SetPower(coll, 6, F.9*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*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 5, 2, 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, 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);