#9058607

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

4570 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)(3,4)(5,7)(6,8)(25,26,30,34)(27,35,36,31)(28,32,38,39)(29,37,40,33);(1,2,6,8)(3,4,5,7)(9,10)(11,17)(12,20)(13,14)(15,18)(16,21)(19,23)(22,24)(25,27,30,36)(26,31,34,35)(28,33,38,37)(29,32,40,39);(3,5)(4,7)(10,14)(11,18)(12,21)(23,24)(25,27,30,36)(26,31,34,35)(28,37,38,33)(29,39,40,32);(1,3,6,5)(2,7,8,4)(9,11)(10,15)(12,22)(13,18)(14,17)(16,24)(19,21)(20,23)(25,27)(26,31)(28,37)(29,32)(30,36)(33,38)(34,35)(39,40);(1,4,6,7)(2,3,8,5)(9,12)(10,16)(11,19)(13,21)(14,20)(15,23)(17,24)(18,22)(25,28,30,38)(26,32,34,39)(27,33,36,37)(29,31,40,35);(1,5,6,3)(2,7,8,4)(9,10)(11,15)(12,16)(13,14)(17,18)(19,23)(20,21)(22,24)(25,29,30,40)(26,33,34,37)(27,32,36,39)(28,31,38,35)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=h,d^2=gh,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,d^c=dgi,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, 8965803);