#8699326

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

4413 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=g,d^2=gh,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,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, 8606522);