#9040279

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

4585 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=g,e^2=g,f^2=g,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, 8947475);