#8798557

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

4347 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)(11,14)(12,15)(13,17)(16,19)(18,22)(20,21)(23,24)(25,26,31,29)(27,34,37,32)(28,38,33,30)(35,40,39,36);(1,2)(3,8)(4,5)(6,7)(9,11,13,19)(10,14,17,16)(12,18,21,24)(15,22,20,23)(26,29)(30,38)(32,34)(36,40);(2,7)(3,4)(9,10,13,17)(11,16,19,14)(12,20,21,15)(18,22,24,23)(25,27,31,37)(26,32,29,34)(28,39,33,35)(30,36,38,40);(1,3,6,4)(2,8,7,5)(9,12,13,21)(10,15,17,20)(11,18,19,24)(14,22,16,23)(25,28)(26,30)(27,35)(29,38)(31,33)(32,40)(34,36)(37,39);(1,4,6,3)(2,5,7,8)(9,10)(11,16)(12,15)(13,17)(14,19)(18,23)(20,21)(22,24)(25,29,31,26)(27,32,37,34)(28,38,33,30)(35,40,39,36);(1,5)(2,3)(4,7)(6,8)(9,11)(10,16)(12,18)(13,19)(14,17)(15,23)(20,22)(21,24)(25,30,31,38)(26,33,29,28)(27,36,37,40)(32,35,34,39)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=h,d^2=i,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8705753);