#512413

order	512
nilpotency class	4
isoclinism	isoclinic class of #511747
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #376 ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

581 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 44 points:

(2,4)(5,6,8,10)(7,11,12,9)(13,14)(15,26)(16,29)(17,22)(18,24)(19,34)(20,21)(23,32)(25,36)(27,37)(28,43)(30,40)(31,42)(33,39)(35,41)(38,44);(5,7,8,12)(6,9,10,11)(13,15,18,20)(14,21,24,26)(16,30,17,27)(19,28,33,36)(22,40,29,37)(23,38,42,41)(25,39,43,34)(31,44,32,35);(1,2,3,4)(13,16,18,17)(14,22,24,29)(15,27,20,30)(19,31,33,32)(21,37,26,40)(23,39,42,34)(25,41,43,38)(28,35,36,44);(13,17,18,16)(14,23,24,42)(15,27,20,30)(19,35,33,44)(21,38,26,41)(22,39,29,34)(25,40,43,37)(28,31,36,32)

Polycyclic presentation:

2^9|a^2=ei,b^2=e,c^2=f,d^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,g^a=gh,h^a=hi,c^b=ci,g^d=gh,h^d=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, 419609);