#9070713

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

4421 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=g,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,e^d=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, 8977909);