#8115499

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

4233 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)(9,10,15,12)(11,16,18,14)(13,20,23,17)(19,22,21,24)(26,30)(27,33)(28,36)(38,40);(2,5)(6,7)(9,11,15,18)(10,16,12,14)(13,21,23,19)(17,22,20,24)(25,26,29,30)(27,34,33,31)(28,32,36,37)(35,40,39,38);(1,2)(3,6)(4,5)(7,8)(10,12)(11,18)(17,20)(19,21)(25,26)(27,31)(28,32)(29,30)(33,34)(35,38)(36,37)(39,40);(1,3)(2,6)(4,8)(5,7)(9,12,15,10)(11,16,18,14)(13,20,23,17)(19,24,21,22)(25,27,29,33)(26,31,30,34)(28,35,36,39)(32,38,37,40);(9,13,15,23)(10,17,12,20)(11,19,18,21)(14,22,16,24)(25,28,29,36)(26,32,30,37)(27,35,33,39)(31,38,34,40);(9,14,15,16)(10,18,12,11)(13,22,23,24)(17,21,20,19)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8022695);