#500440

order	512
nilpotency class	4
isoclinism	isoclinic class of #499712
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₄	D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

807 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)(7,11)(8,9)(10,12);(6,9)(10,11)(13,14,19,23)(15,24,27,43)(16,21,18,33)(17,29,32,41)(20,35,38,26)(22,28,40,31)(25,42,34,39)(30,36,44,37);(2,4)(13,15,19,27)(14,20,23,38)(16,25,18,34)(17,30,32,44)(21,36,33,37)(22,39,40,42)(24,41,43,29)(26,28,35,31);(5,7,8,12)(6,10,9,11)(13,16,19,18)(14,21,23,33)(15,25,27,34)(17,28,32,31)(20,36,38,37)(22,29,40,41)(24,42,43,39)(26,30,35,44)

Polycyclic presentation:

2^9|b^2=i,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,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, 407636);