#9075947

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

4287 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=gh,d^2=gh,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,e^d=egi,f^d=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, 8983143);