#512207

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	#33179	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

666 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)(18,31)(21,32)(23,34)(24,35)(25,36)(28,29)(30,37)(33,38)(39,43)(40,42)(41,44);(6,9)(10,12)(13,14,21,36,18,25,16,20,19,23,17,24,31,35,32,34)(15,27,37,38,29,43,28,42,30,22,26,40,44,39,41,33);(5,7,8,11)(6,10,9,12)(13,15,16,28,31,44,21,37,19,30,32,41,18,29,17,26)(14,22,20,33,35,43,36,40,23,27,34,42,25,39,24,38);(2,4)(14,23)(20,34)(22,27)(24,36)(25,35)(33,42)(38,40)(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,c^b=ci,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, 419403);