#8755435

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

4357 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=hi,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, 8662631);