#8471819

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

4570 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8379015);