#9206533

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

4344 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)(6,7)(9,10,14,15)(11,17,18,12)(13,21,23,16)(19,20,24,22)(26,30)(27,35)(28,38)(34,40);(1,2,5,4)(3,7,8,6)(9,10,14,15)(11,12,18,17)(13,16,23,21)(19,20,24,22)(25,26,31,30)(27,36,35,32)(28,33,38,39)(29,40,37,34);(1,3,5,8)(2,6,4,7)(10,15)(11,18)(16,21)(19,24)(25,27,31,35)(26,32,30,36)(28,37,38,29)(33,40,39,34);(1,4,5,2)(3,7,8,6)(9,11,14,18)(10,12,15,17)(13,19,23,24)(16,20,21,22)(25,28)(26,33)(27,37)(29,35)(30,39)(31,38)(32,40)(34,36);(9,12)(10,11)(13,22)(14,17)(15,18)(16,24)(19,21)(20,23)(25,29,31,37)(26,34,30,40)(27,28,35,38)(32,33,36,39);(9,13,14,23)(10,16,15,21)(11,19,18,24)(12,20,17,22)(25,30,31,26)(27,36,35,32)(28,39,38,33)(29,40,37,34)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=g,d^2=gh,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,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, 9113729);