#8095792

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

4665 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=i,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi

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, 8002988);