#8700382

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

4439 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)(3,7)(4,5)(6,8)(9,10,15,14)(11,18,12,16)(13,21,22,17)(19,23,20,24)(25,26)(27,38)(28,34)(29,39)(30,35)(31,36)(32,37)(33,40);(2,5)(6,7)(9,10,15,14)(11,18,12,16)(13,17,22,21)(19,24,20,23)(25,27)(26,33)(28,35)(29,31)(30,39)(32,40)(34,36)(37,38);(1,2)(3,6)(4,5)(7,8)(9,11,15,12)(10,16,14,18)(13,19,22,20)(17,23,21,24)(25,28,32,29)(26,34,37,39)(27,35,40,31)(30,33,36,38);(1,3,4,8)(2,6,5,7)(9,12,15,11)(10,16,14,18)(13,20,22,19)(17,23,21,24)(25,29,32,28)(26,34,37,39)(27,35,40,31)(30,38,36,33);(9,13,15,22)(10,17,14,21)(11,19,12,20)(16,23,18,24)(25,30)(26,35)(27,39)(28,33)(29,38)(31,37)(32,36)(34,40);(9,14,15,10)(11,18,12,16)(13,17,22,21)(19,23,20,24)(25,31,32,35)(26,36,37,30)(27,29,40,28)(33,34,38,39)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gi,d^2=gh,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,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, 8607578);