#9723165

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	#48476	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4226 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=gi,d^2=g,e^2=h,f^2=gi,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, 9630361);