#9027306

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

4440 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:

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

Polycyclic presentation:

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

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, 8934502);