#6631892

order	512
nilpotency class	3
isoclinism	isoclinic class of #6631116
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#35276	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1475 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dh,c^b=ch,d^b=di,e^b=ei,d^c=dfh,g^c=gi,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6539088);