#8322129

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

4434 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,d^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8229325);