#8696766

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

4589 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=gi,d^2=gh,f^2=gh,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, 8603962);