#9681750

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

4480 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)(11,13)(12,16)(14,15)(17,21)(18,20)(19,22)(23,24)(25,26,31,33)(27,34,29,32)(28,37,39,30)(35,36,38,40);(1,2)(3,8)(4,6)(5,7)(12,19)(16,22)(18,23)(20,24)(25,27,31,29)(26,32,33,34)(28,35,39,38)(30,36,37,40);(1,3,5,6)(2,4,7,8)(10,15)(11,17)(16,22)(18,23);(9,11,14,17)(10,13,15,21)(12,18,19,23)(16,20,22,24)(25,28)(26,30)(27,35)(29,38)(31,39)(32,36)(33,37)(34,40);(1,2)(3,4)(5,7)(6,8)(9,12)(10,16)(11,18)(13,20)(14,19)(15,22)(17,23)(21,24)(25,29,31,27)(26,32,33,34)(28,38,39,35)(30,36,37,40);(1,4)(2,6)(3,7)(5,8)(9,13)(10,17)(11,15)(12,20)(14,21)(16,23)(18,22)(19,24)(25,30)(26,28)(27,36)(29,40)(31,37)(32,35)(33,39)(34,38)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9588946);