#8696636

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

4424 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)(3,7)(9,10)(11,17)(12,13)(14,18)(15,16)(19,22)(20,21)(23,24)(25,26,29,32)(27,30,34,37)(28,31,35,38)(33,36,39,40);(1,2,4,5)(3,6,7,8)(10,16)(12,17)(18,24)(20,22)(25,27,29,34)(26,30,32,37)(28,33,35,39)(31,36,38,40);(1,2)(3,6)(4,5)(7,8)(9,11,15,13)(10,17,16,12)(14,19,23,21)(18,22,24,20)(25,26)(27,30)(28,31)(29,32)(33,36)(34,37)(35,38)(39,40);(1,2)(3,6)(4,5)(7,8)(9,12)(10,11)(13,16)(14,22)(15,17)(18,21)(19,24)(20,23)(25,27)(26,30)(28,33)(29,34)(31,36)(32,37)(35,39)(38,40);(1,3,4,7)(2,6,5,8)(9,13,15,11)(10,12,16,17)(14,19,23,21)(18,22,24,20)(25,26)(27,30)(28,31)(29,32)(33,36)(34,37)(35,38)(39,40);(9,14,15,23)(10,18,16,24)(11,19,13,21)(12,20,17,22)(25,28,29,35)(26,31,32,38)(27,33,34,39)(30,36,37,40)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=gi,e^2=g,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, 8603832);