#9224377

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

4636 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=h,d^2=i,e^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, 9131573);