#8093368

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

4498 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=h,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^b=eghi

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, 8000564);