#8622158

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

4522 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=h,e^2=gi,f^2=h,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, 8529354);