#512199

order	512
nilpotency class	4
isoclinism	isoclinic class of #511951
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#466	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

637 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)(7,10)(8,9)(11,12)(14,20)(15,26)(16,17)(18,30)(21,31)(23,33)(24,34)(25,35)(27,28)(29,36)(32,38)(39,43)(40,41)(42,44);(6,9)(10,12)(13,14,21,35,18,25,16,20,19,23,17,24,30,34,31,33)(15,22,36,44,28,39,27,32,29,37,26,38,41,43,40,42);(5,7,8,11)(6,10,9,12)(13,15,16,27,30,41,21,36,19,29,31,40,18,28,17,26)(14,22,20,32,34,43,35,44,23,37,33,42,25,39,24,38);(2,4)(14,23)(20,33)(22,37)(24,35)(25,34)(32,42)(38,44)(39,43)

Polycyclic presentation:

2^9|b^2=fi,c^2=e,e^2=hi,f^2=h,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fhi,h^a=hi,d^b=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419395);