#8697531

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

4700 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=hi,d^2=hi,f^2=g,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, 8604727);