#8093273

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

4642 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=g,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^b=eghi

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, 8000469);