#511902

order	512
nilpotency class	4
isoclinism	isoclinic class of #511755
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#578	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1044 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=e,d^2=i,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fh,g^a=gi,h^a=hi,f^c=fh,h^c=hi,g^d=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419098);