#512291

order	512
nilpotency class	4
isoclinism	isoclinic class of #511747
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#466	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

644 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 44 points:

(2,3)(5,6)(7,9)(8,10)(11,12)(13,14)(15,24)(16,21)(17,20)(18,31)(19,25)(22,30)(23,29)(26,37)(27,34)(28,35)(32,40)(33,44)(36,43)(38,42)(39,41);(1,2,4,3)(5,7,8,11)(6,9,10,12)(15,25)(19,24)(26,40)(27,41)(32,37)(33,38)(34,39)(42,44);(6,10)(9,12)(13,15,16,25)(14,19,21,24)(17,26,28,40)(18,27,29,41)(20,32,35,37)(22,33,36,38)(23,34,31,39)(30,42,43,44);(1,3,4,2)(14,20)(17,30)(18,29)(19,32)(21,35)(22,23)(24,37)(26,42)(27,41)(28,43)(31,36)(33,34)(38,39)(40,44)

Polycyclic presentation:

2^9|b^2=efi,c^2=i,d^2=e,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,g^a=gh,h^a=hi,c^b=ci,g^d=gh,h^d=hi

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, 419487);