#8724835

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) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,d^2=hi,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,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, 8632031);