#8926069

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

4520 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=g,d^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,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, 8833265);