#501554

order	512
nilpotency class	4
isoclinism	isoclinic class of #499724
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

551 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)(13,14,16,19)(15,23,25,20)(17,21,27,32)(18,22,28,33)(24,34,42,35)(26,40,31,36)(29,39,41,37)(30,44,43,38);(6,9)(10,11)(14,19)(15,24)(17,29)(18,28)(20,34)(21,37)(23,35)(25,42)(26,43)(27,41)(30,31)(32,39)(36,38)(40,44);(2,4)(13,15,16,25)(14,20,19,23)(17,30,27,43)(18,31,28,26)(21,38,32,44)(22,40,33,36)(24,41,42,29)(34,39,35,37);(5,7,8,12)(6,10,9,11)(14,19)(15,25)(21,32)(22,33)(24,42)(26,31)(30,43)(37,39)

Polycyclic presentation:

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

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, 408750);