#8587663

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

4391 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)(3,8)(4,6)(5,7)(9,10,15,18)(11,16,19,13)(12,14,17,23)(20,21,24,22)(27,33)(30,34)(35,39)(38,40);(2,4)(7,8)(9,11)(10,16)(12,20)(13,18)(14,21)(15,19)(17,24)(22,23)(25,26,29,32)(27,34,33,30)(28,31,37,36)(35,40,39,38);(1,3,6,5)(2,7,4,8)(11,19)(13,16)(20,24)(21,22)(25,27,29,33)(26,30,32,34)(28,35,37,39)(31,38,36,40);(1,2,6,4)(3,7,5,8)(9,12)(10,14)(11,20)(13,22)(15,17)(16,21)(18,23)(19,24)(25,26)(27,30)(28,36)(29,32)(31,37)(33,34)(35,40)(38,39);(1,4,6,2)(3,7,5,8)(9,13)(10,11)(12,22)(14,20)(15,16)(17,21)(18,19)(23,24)(25,27)(26,30)(28,35)(29,33)(31,38)(32,34)(36,40)(37,39);(1,5,6,3)(2,7,4,8)(9,14)(10,17)(11,21)(12,18)(13,20)(15,23)(16,24)(19,22)(25,28)(26,31)(27,35)(29,37)(30,38)(32,36)(33,39)(34,40)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=h,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=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, 8494859);