#8701441

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4381 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,d^2=gi,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8608637);