#9224254

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

4489 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9131450);