#9040987

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

4509 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,d^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8948183);