#8417273

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

4232 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)(9,10)(11,16)(12,14)(13,21)(15,17)(18,24)(19,20)(22,23)(26,30)(31,36)(32,37)(38,40);(2,5)(6,7)(9,11,15,14)(10,16,17,12)(13,22,24,19)(18,20,21,23)(27,33)(31,36)(34,39)(38,40);(1,2)(3,6)(4,5)(7,8)(10,17)(11,14)(18,21)(19,22)(25,26)(27,31)(28,32)(29,30)(33,36)(34,38)(35,37)(39,40);(1,3)(2,6)(4,8)(5,7)(9,12)(10,11)(13,23)(14,17)(15,16)(18,22)(19,21)(20,24)(25,27,29,33)(26,31,30,36)(28,34,35,39)(32,38,37,40);(9,13,15,24)(10,18,17,21)(11,19,14,22)(12,20,16,23)(25,26)(27,31)(28,32)(29,30)(33,36)(34,38)(35,37)(39,40);(9,14,15,11)(10,12,17,16)(13,22,24,19)(18,20,21,23)(25,28,29,35)(26,32,30,37)(27,34,33,39)(31,38,36,40)

Polycyclic presentation:

2^9|b^2=gi,d^2=ghi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,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, 8324469);