#8634818

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

4439 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,d^2=h,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=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, 8542014);