#506312

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	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

548 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)(13,14,16,20)(15,23,26,19)(17,21,24,31)(18,22,29,35)(25,37,42,32)(27,38,40,33)(28,39,43,34)(30,36,41,44);(6,9)(10,11)(13,15,24,40,29,43,30,25,16,26,17,27,18,28,41,42)(14,19,31,38,35,39,36,32,20,23,21,33,22,34,44,37);(2,4)(15,25)(17,30)(18,29)(19,32)(21,36)(22,35)(23,37)(24,41)(26,42)(27,28)(31,44)(33,34)(38,39)(40,43);(5,7,8,12)(6,10,9,11)(14,20)(19,23)(21,31)(22,35)(32,37)(33,38)(34,39)(36,44)

Polycyclic presentation:

2^9|a^2=e,b^2=gi,d^2=ei,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, 413508);