#8695394

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

4533 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=gi,f^2=g,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, 8602590);