#8648967

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

4434 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8556163);