#9075949

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

4542 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)(5,8)(6,7)(9,10,15,19)(11,16,20,24)(12,14,18,23)(13,17,21,22)(25,26,30,33)(27,34,29,31)(28,32,36,37)(35,40,38,39);(1,3,6,5)(2,4,7,8)(9,11)(10,16)(12,21)(13,18)(14,22)(15,20)(17,23)(19,24)(25,27,30,29)(26,31,33,34)(28,35,36,38)(32,39,37,40);(1,4,6,8)(2,5,7,3)(11,20)(13,21)(16,24)(17,22)(28,36)(32,37)(35,38)(39,40);(1,4,6,8)(2,3,7,5)(9,12)(10,14)(11,21)(13,20)(15,18)(16,22)(17,24)(19,23)(25,26)(27,31)(28,37)(29,34)(30,33)(32,36)(35,40)(38,39);(1,4)(2,3)(5,7)(6,8)(9,13,15,21)(10,17,19,22)(11,18,20,12)(14,16,23,24)(25,28)(26,32)(27,35)(29,38)(30,36)(31,39)(33,37)(34,40);(1,5,6,3)(2,4,7,8)(9,14)(10,18)(11,22)(12,19)(13,16)(15,23)(17,20)(21,24)(25,29,30,27)(26,31,33,34)(28,38,36,35)(32,39,37,40)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=h,d^2=h,e^2=hi,f^2=gh,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, 8983145);