#9730641

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

4283 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=hi,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,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, 9637837);