#9368174

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

4888 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)(3,7)(9,10)(11,15)(12,19)(13,14)(16,20)(17,21)(18,24)(22,23)(25,26)(27,33)(28,36)(29,30)(31,34)(32,38)(35,39)(37,40);(2,5)(3,7)(11,17)(15,21)(18,23)(22,24)(26,30)(31,33)(32,36)(39,40);(1,2)(3,6)(4,5)(7,8)(10,14)(15,21)(16,19)(22,24)(27,34)(31,33)(35,37)(39,40);(3,7)(6,8)(9,10)(11,15)(12,16)(13,14)(17,21)(18,22)(19,20)(23,24)(25,27)(26,31)(28,37)(29,34)(30,33)(32,40)(35,38)(36,39);(1,3)(2,6)(4,7)(5,8)(9,11)(10,15)(12,18)(13,17)(14,21)(16,22)(19,24)(20,23)(25,28,29,38)(26,32,30,36)(27,35,34,37)(31,39,33,40);(1,2)(3,6)(4,5)(7,8)(9,12)(10,16)(11,18)(13,20)(14,19)(15,22)(17,23)(21,24)(25,26)(27,31)(28,32)(29,30)(33,34)(35,39)(36,38)(37,40)

Polycyclic presentation:

2^9|e^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=dghi,e^d=egi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9275370);