#9625443

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

4229 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,e^d=egi,f^d=fgi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9532639);