#9105321

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

4457 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:

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

Polycyclic presentation:

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

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, 9012517);