#9224253

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

4639 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)(9,10,15,16)(11,18,12,17)(13,19,20,14)(21,22,23,24)(25,26)(27,29)(28,36)(30,31)(32,34)(33,38)(35,39)(37,40);(3,7)(5,8)(10,16)(13,20)(17,18)(21,23)(26,31)(27,34)(33,36)(35,40);(1,2)(3,5)(4,6)(7,8)(9,11,15,12)(10,17,16,18)(13,21,20,23)(14,22,19,24)(25,27,30,34)(26,32,31,29)(28,35,38,40)(33,39,36,37);(9,12,15,11)(10,18,16,17)(13,21,20,23)(14,22,19,24)(25,28)(26,33)(27,35)(29,37)(30,38)(31,36)(32,39)(34,40);(1,3)(2,5)(4,7)(6,8)(9,13)(10,19)(11,21)(12,23)(14,16)(15,20)(17,24)(18,22)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,38)(37,40);(1,2)(3,5)(4,6)(7,8)(9,14,15,19)(10,20,16,13)(11,22,12,24)(17,23,18,21)(25,29,30,32)(26,34,31,27)(28,37,38,39)(33,40,36,35)

Polycyclic presentation:

2^9|a^2=hi,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, 9131449);