#502095

order	512
nilpotency class	4
isoclinism	isoclinic class of #499769
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#578	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

746 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 68 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=i,e^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,e^b=ei,g^b=gh,h^b=hi,g^c=gh,h^c=hi,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 409291);