#7713885

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

4578 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)(9,10)(11,18)(12,14)(13,21)(15,16)(17,22)(19,23)(20,24)(28,36)(32,37)(34,40)(38,39);(1,2,4,5)(3,6,7,8)(9,11)(10,14)(12,15)(13,19)(16,18)(17,23)(20,22)(21,24)(25,26)(27,33)(28,32)(29,30)(31,35)(34,39)(36,37)(38,40);(10,16)(11,12)(13,22)(23,24)(26,30)(31,33)(32,37)(38,39);(9,12,15,11)(10,14,16,18)(13,20,22,19)(17,23,21,24)(25,27,29,35)(26,31,30,33)(28,34,36,40)(32,38,37,39);(1,3,4,7)(2,6,5,8)(9,13,15,22)(10,17,16,21)(11,19,12,20)(14,23,18,24)(25,26)(27,31)(28,32)(29,30)(33,35)(34,38)(36,37)(39,40);(9,14)(10,11)(12,16)(13,23)(15,18)(17,19)(20,21)(22,24)(25,28)(26,32)(27,34)(29,36)(30,37)(31,38)(33,39)(35,40)

Polycyclic presentation:

2^9|b^2=gh,d^2=i,e^2=g,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^b=di,d^c=dh,e^c=ei

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, 7621081);