#9159897

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

4586 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=hi,d^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9067093);