#8183756

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

4622 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=g,d^2=gh,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

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, 8090952);