#506192

order	512
nilpotency class	4
isoclinism	isoclinic class of #500834
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

711 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)(14,19)(20,32)(21,33)(22,34)(23,39)(35,36)(37,44)(38,40);(5,7,8,12)(6,9,10,11)(13,14,18,22,28,33,43,44,17,19,30,34,16,21,29,37)(15,23,26,35,24,40,41,32,25,39,42,36,27,38,31,20);(2,4)(13,15)(14,20)(16,24)(17,25)(18,31)(19,32)(21,35)(22,38)(23,37)(26,29)(27,28)(30,41)(33,36)(34,40)(39,44)(42,43);(6,10)(9,11)(13,16,17,28)(14,21,19,33)(15,24,25,27)(18,29,30,43)(20,35,32,36)(22,37,34,44)(23,40,39,38)(26,41,42,31)

Polycyclic presentation:

2^9|b^2=egi,d^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,g^c=gh,h^c=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, 413388);