#8924898

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

4469 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8832094);