#8718951

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

4540 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=hi,d^2=i,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8626147);