#8183764

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

4532 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,d^2=h,e^2=h,f^2=g,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, 8090960);