#8874928

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

4358 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=g,d^2=g,e^2=g,f^2=i,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, 8782124);