#72343380

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#442
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₂²×C₈	C₂⁵
prime radical	#7622812	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ #20385 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

4779 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:

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

Polycyclic presentation:

2^10|a^2=f,b^2=fhij,c^2=fhj,f^2=i,h^2=i,i^2=j,b^a=bh,c^a=ci,c^b=ci,d^a=dj,d^b=dj,d^c=dg,e^c=ej,f^b=fi,f^c=fj,h^a=hj,h^b=hi,h^c=hj,i^b=ij

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343381, #72343421, #72343897
maximal normal subgroup of	#72343381, #72343421, #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.6*F.8*F.9*F.10);;
gap> SetPower(coll, 3, F.6*F.8*F.10);;
gap> SetPower(coll, 6, F.9);;
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.9);;
gap> SetConjugate(coll, 4, 1, F.4*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.7);;
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, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
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);