#8696718

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

4715 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=ghi,f^2=h,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, 8603914);