#9723146

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
prime radical	#51745	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4343 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)(11,21)(12,17)(13,18)(14,19)(15,20)(16,23)(22,24)(25,26,31,36)(27,32,37,40)(28,38,39,33)(29,34,35,30);(1,2,4,6)(3,5,8,7)(9,11,15,23)(10,16,20,21)(12,22,13,14)(17,19,18,24)(25,27,31,37)(26,32,36,40)(28,34,39,30)(29,38,35,33);(1,2)(3,5)(4,6)(7,8)(9,12,15,13)(10,17,20,18)(11,22,23,14)(16,19,21,24)(25,28,31,39)(26,33,36,38)(27,30,37,34)(29,32,35,40);(1,3,4,8)(2,5,6,7)(9,13,15,12)(10,18,20,17)(11,14,23,22)(16,24,21,19)(25,28)(26,33)(27,34)(29,32)(30,37)(31,39)(35,40)(36,38);(9,14)(10,19)(11,12)(13,23)(15,22)(16,18)(17,21)(20,24)(25,29)(26,34)(27,33)(28,32)(30,36)(31,35)(37,38)(39,40);(9,10)(11,16)(12,17)(13,18)(14,24)(15,20)(19,22)(21,23)(25,30,31,34)(26,35,36,29)(27,28,37,39)(32,33,40,38)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=gi,d^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,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, 9630342);