#9835836

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

4226 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=hi,d^2=ghi,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,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, 9743032);