#502507

order	512
nilpotency class	4
isoclinism	isoclinic class of #500854
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×#41	C₂×D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

643 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)(7,20)(8,27)(9,32)(10,38)(11,18)(12,19)(13,42)(14,24)(15,23)(16,34)(17,33)(21,54)(22,52)(25,56)(26,57)(28,65)(29,48)(30,67)(31,45)(35,49)(36,44)(37,68)(39,43)(40,58)(41,51)(46,63)(47,61)(50,59)(53,60)(55,62)(64,66);(1,3,4,2)(5,7,21,29,13,26,10,24,11,25,33,61,12,40,39,63)(6,14,43,57,19,48,17,20,18,46,38,58,42,47,54,56)(8,28,59,36,31,22,30,9,23,60,66,37,41,34,62,35)(15,49,67,16,45,68,50,53,27,32,55,52,51,44,64,65);(2,3)(5,8)(6,15)(7,22)(9,29)(10,30)(11,23)(12,41)(13,31)(14,44)(16,20)(17,50)(18,27)(19,45)(21,59)(24,37)(25,34)(26,60)(28,40)(32,47)(33,66)(35,61)(36,63)(38,55)(39,62)(42,51)(43,67)(46,68)(48,49)(52,56)(53,58)(54,64)(57,65);(5,9,33,28,13,37,39,22,11,35,21,60,12,36,10,34)(6,16,38,44,19,53,54,49,18,52,43,68,42,65,17,32)(7,23,61,59,26,41,63,30,25,8,29,66,40,31,24,62)(14,45,58,64,48,27,56,67,46,51,57,50,47,15,20,55)

Polycyclic presentation:

2^9|a^2=fi,b^2=efh,d^2=eh,e^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,g^b=gi,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 409703);