#501536

order	512
nilpotency class	4
isoclinism	isoclinic class of #499712
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	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄ ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

558 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408732);