#9040907

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

4547 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)(3,6)(4,5)(7,8)(25,26,31,33)(27,32,30,40)(28,29,34,38)(35,39,37,36);(1,3)(2,4)(5,7)(6,8)(9,10)(11,17)(12,20)(13,14)(15,19)(16,21)(18,23)(22,24)(25,27,31,30)(26,32,33,40)(28,37,34,35)(29,36,38,39);(1,3)(2,6)(4,8)(5,7)(10,14)(12,21)(15,17)(18,22)(26,33)(27,30)(29,38)(35,37);(1,4,7,6)(2,3,8,5)(9,11)(10,15)(12,22)(13,19)(14,17)(16,24)(18,21)(20,23)(25,28)(26,29)(27,35)(30,37)(31,34)(32,39)(33,38)(36,40);(1,5,7,3)(2,4,8,6)(9,12)(10,16)(11,18)(13,21)(14,20)(15,23)(17,24)(19,22)(25,29)(26,34)(27,36)(28,33)(30,39)(31,38)(32,35)(37,40);(1,6,7,4)(2,3,8,5)(9,10)(11,15)(12,16)(13,14)(17,19)(18,23)(20,21)(22,24)(25,30,31,27)(26,32,33,40)(28,37,34,35)(29,39,38,36)

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=g,e^2=g,f^2=gh,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, 8948103);