#10166297

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

4321 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:

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

Polycyclic presentation:

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

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, 10073493);