#9065305

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

4396 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=g,f^2=gh,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, 8972501);