#8692581

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

4499 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,d^2=gh,e^2=gh,f^2=h,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, 8599777);