#9224388

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

4557 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)(3,4)(5,7)(6,8)(9,10,15,17)(11,16,20,24)(12,21,13,14)(18,19,22,23)(25,26)(27,35)(28,38)(29,40)(30,31)(32,37)(33,39)(34,36);(1,3)(2,6)(4,7)(5,8)(9,11,15,20)(10,16,17,24)(12,22,13,18)(14,19,21,23)(25,27,31,37)(26,32,30,35)(28,36,39,29)(33,34,38,40);(9,12,15,13)(10,14,17,21)(11,18,20,22)(16,23,24,19)(25,28,31,39)(26,33,30,38)(27,36,37,29)(32,34,35,40);(2,7)(4,6)(9,13,15,12)(10,14,17,21)(11,18,20,22)(16,19,24,23)(25,29)(26,34)(27,28)(30,40)(31,36)(32,38)(33,35)(37,39);(1,2)(3,6)(4,8)(5,7)(9,11)(10,16)(12,18)(13,22)(14,23)(15,20)(17,24)(19,21)(25,27)(26,35)(28,36)(29,39)(30,32)(31,37)(33,40)(34,38);(1,4,5,6)(2,3,7,8)(9,14)(10,13)(11,19)(12,17)(15,21)(16,18)(20,23)(22,24)(25,30,31,26)(27,35,37,32)(28,38,39,33)(29,34,36,40)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=h,d^2=i,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, 9131584);