#8993591

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

4578 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)(3,7)(4,5)(6,8)(9,10,15,20)(11,21,22,16)(12,13,18,17)(14,19,23,24)(25,26)(27,34)(28,36)(29,33)(30,31)(32,37)(35,39)(38,40);(3,8)(6,7)(9,11,15,22)(10,16,20,21)(12,19,18,24)(13,14,17,23)(25,27,31,33)(26,29,30,34)(28,35,37,40)(32,39,36,38);(1,3,5,8)(2,6,4,7)(12,18)(13,17)(14,23)(19,24)(26,30)(28,37)(29,34)(35,40);(3,8)(6,7)(9,12)(10,17)(11,19)(13,20)(14,21)(15,18)(16,23)(22,24)(25,28,31,37)(26,32,30,36)(27,35,33,40)(29,39,34,38);(1,3)(2,7)(4,6)(5,8)(9,13,15,17)(10,18,20,12)(11,14,22,23)(16,24,21,19)(25,29,31,34)(26,33,30,27)(28,38,37,39)(32,35,36,40);(1,4,5,2)(3,6,8,7)(9,14)(10,19)(11,17)(12,21)(13,22)(15,23)(16,18)(20,24)(25,30,31,26)(27,34,33,29)(28,36,37,32)(35,38,40,39)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=i,d^2=hi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=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, 8900787);