#9009252

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

4542 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)(3,8)(4,7)(5,6)(9,10)(11,17)(12,20)(13,14)(15,18)(16,21)(19,22)(23,24);(3,4)(7,8)(10,14)(15,17)(16,20)(22,23)(25,26)(27,31)(28,32)(29,30)(33,37)(34,39)(35,38)(36,40);(1,3)(2,7)(4,6)(5,8)(10,14)(11,18)(12,21)(22,23)(26,30)(27,33)(28,35)(39,40);(3,4)(7,8)(9,11,13,18)(10,15,14,17)(12,19,21,24)(16,22,20,23)(25,27,29,33)(26,31,30,37)(28,36,35,34)(32,40,38,39);(1,4,6,3)(2,7,5,8)(9,10)(11,17)(12,16)(13,14)(15,18)(19,23)(20,21)(22,24)(25,28,29,35)(26,32,30,38)(27,34,33,36)(31,39,37,40);(1,5,6,2)(3,7,4,8)(9,12,13,21)(10,16,14,20)(11,19,18,24)(15,22,17,23)(25,26)(27,31)(28,32)(29,30)(33,37)(34,39)(35,38)(36,40)

Polycyclic presentation:

2^9|d^2=hi,e^2=i,f^2=h,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, 8916448);