#9029047

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

4285 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)(11,16,13,19)(12,18,23,22)(14,21,24,20)(26,28)(27,35)(30,40)(33,37);(1,2,5,4)(3,7,8,6)(9,11)(10,16)(12,21)(13,15)(14,22)(17,19)(18,24)(20,23)(25,26,31,28)(27,36,35,32)(29,37,39,33)(30,34,40,38);(1,3,5,8)(2,6,4,7)(10,17)(11,13)(12,23)(14,24)(25,27)(26,32)(28,36)(29,30)(31,35)(33,34)(37,38)(39,40);(3,8)(6,7)(9,12,15,23)(10,18,17,22)(11,21,13,20)(14,16,24,19)(25,28,31,26)(27,36,35,32)(29,37,39,33)(30,38,40,34);(1,3)(2,7)(4,6)(5,8)(9,13,15,11)(10,19,17,16)(12,21,23,20)(14,18,24,22)(25,29)(26,33)(27,30)(28,37)(31,39)(32,34)(35,40)(36,38);(1,4,5,2)(3,6,8,7)(9,14,15,24)(10,20,17,21)(11,22,13,18)(12,16,23,19)(25,30)(26,34)(27,29)(28,38)(31,40)(32,33)(35,39)(36,37)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8936243);