#9064953

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

4307 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gh,d^2=gh,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,e^d=egi,f^d=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, 8972149);