#9836785

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

4297 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=g,d^2=g,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,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, 9743981);