#9229479

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

4580 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gh,d^2=gh,e^2=ghi,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, 9136675);