#8222655

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

4454 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:

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

Polycyclic presentation:

2^9|b^2=gh,d^2=gi,e^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi

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, 8129851);