#506183

order	512
nilpotency class	4
isoclinism	isoclinic class of #500829
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	#24266	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

724 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)(8,10)(9,12);(5,7,8,12)(6,9,10,11)(13,14,17,21,29,36,43,44,18,22,31,38,16,20,28,35)(15,23,25,32,24,39,41,34,27,40,42,33,26,37,30,19);(2,4)(13,15)(14,19)(16,24)(17,30)(18,27)(20,32)(21,37)(22,34)(23,35)(25,28)(26,29)(31,41)(33,36)(38,39)(40,44)(42,43);(6,10)(9,11)(13,16,18,29)(14,20,22,36)(15,24,27,26)(17,28,31,43)(19,32,34,33)(21,35,38,44)(23,39,40,37)(25,41,42,30)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 413379);