#9390445

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=ghi,d^2=h,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dghi,e^d=egi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9297641);