#8459114

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

4544 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,d^2=h,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=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, 8366310);