#9003143

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

4375 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)(6,8)(25,26,32,31)(27,35,29,33)(28,34,38,40)(30,36,37,39);(3,7)(6,8)(9,10,14,12)(11,15,17,19)(13,16,21,20)(18,22,23,24)(25,27,32,29)(26,33,31,35)(28,37,38,30)(34,36,40,39);(1,2)(3,6)(4,5)(7,8)(10,12)(11,17)(13,21)(22,24)(25,28,32,38)(26,34,31,40)(27,37,29,30)(33,36,35,39);(9,11)(10,15)(12,19)(13,18)(14,17)(16,22)(20,24)(21,23)(25,29,32,27)(26,35,31,33)(28,37,38,30)(34,36,40,39);(1,3)(2,6)(4,7)(5,8)(9,12,14,10)(11,15,17,19)(13,20,21,16)(18,22,23,24)(25,30,32,37)(26,36,31,39)(27,28,29,38)(33,40,35,34);(1,2)(3,6)(4,5)(7,8)(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,31,32,26)(27,35,29,33)(28,34,38,40)(30,39,37,36)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8910339);