#8393424

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

4491 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,d^2=i,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8300620);