#8425269

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

4625 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=gi,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,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, 8332465);