#8839312

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

4579 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8746508);