#9040456

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

4605 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:

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

Polycyclic presentation:

2^9|a^2=gh,d^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, 8947652);