#8755190

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

4353 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=gi,d^2=ghi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8662386);