#72343368

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	C₂³×#106
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₂²×C₈	C₂⁵
prime radical	C₂×#525	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#525 ◁ C₂×#2929 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3639 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

148 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343381, #72343421, #72343764
maximal normal subgroup of	#72343381, #72343421, #72343764

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.9);;
gap> SetPower(coll, 6, 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.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.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);