#9075943

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

4399 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)(27,33)(28,30)(29,32)(31,35)(34,39)(36,37)(38,40);(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)(26,31)(28,33)(29,34)(30,35)(32,38)(36,39)(37,40);(1,4,6,8)(2,5,7,3)(11,20)(13,21)(16,24)(17,22)(29,37)(32,36)(34,40)(38,39);(1,4,6,8)(2,3,7,5)(9,12)(10,14)(11,21)(13,20)(15,18)(16,22)(17,24)(19,23)(25,28,30,26)(27,33,35,31)(29,32,37,36)(34,38,40,39);(1,4)(2,3)(5,7)(6,8)(9,13,15,21)(10,17,19,22)(11,18,20,12)(14,16,23,24)(25,29,30,37)(26,32,28,36)(27,34,35,40)(31,38,33,39);(1,5,6,3)(2,4,7,8)(9,14)(10,18)(11,22)(12,19)(13,16)(15,23)(17,20)(21,24)(25,27)(26,33)(28,31)(29,34)(30,35)(32,39)(36,38)(37,40)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=h,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, 8983139);