#8855457

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

4470 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)(3,7)(4,5)(6,8)(9,10)(11,17)(12,20)(13,14)(15,18)(16,22)(19,23)(21,24)(25,26,29,33)(27,31,35,30)(28,36,37,32)(34,38,40,39);(2,5)(6,7)(10,14)(15,17)(16,20)(23,24)(25,26,29,33)(27,31,35,30)(28,36,37,32)(34,38,40,39);(1,2)(3,6)(4,5)(7,8)(11,18)(15,17)(19,21)(23,24)(25,27,29,35)(26,30,33,31)(28,34,37,40)(32,38,36,39);(1,3)(2,6)(4,8)(5,7)(9,11)(10,15)(12,21)(13,18)(14,17)(16,24)(19,22)(20,23)(25,27)(26,31)(28,34)(29,35)(30,33)(32,39)(36,38)(37,40);(9,12,13,22)(10,16,14,20)(11,19,18,21)(15,23,17,24)(25,26)(27,30)(28,32)(29,33)(31,35)(34,38)(36,37)(39,40);(9,10)(11,15)(12,16)(13,14)(17,18)(19,23)(20,22)(21,24)(25,28)(26,32)(27,34)(29,37)(30,38)(31,39)(33,36)(35,40)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8762653);