#9236435

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

4442 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=ghi,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,e^d=egi,f^d=fgi,f^e=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, 9143631);