#9304662

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

4357 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9211858);