#9724834

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

4543 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)(11,16)(12,18)(13,19)(14,24)(15,17)(20,21)(22,23)(25,26)(27,33)(28,39)(29,32)(30,34)(31,37)(35,40)(36,38);(1,2,5,4)(3,6,8,7)(9,11,15,14)(10,16,17,24)(12,20,19,22)(13,23,18,21)(25,27,32,38)(26,33,29,36)(28,37,30,35)(31,34,40,39);(1,2,5,4)(3,7,8,6)(10,17)(12,19)(16,24)(20,22)(25,28,32,30)(26,34,29,39)(27,35,38,37)(31,36,40,33);(1,3)(2,6)(4,7)(5,8)(9,12,15,19)(10,18,17,13)(11,20,14,22)(16,21,24,23)(25,29,32,26)(27,36,38,33)(28,34,30,39)(31,37,40,35);(1,3)(2,7)(4,6)(5,8)(9,13,15,18)(10,19,17,12)(11,21,14,23)(16,20,24,22)(25,30,32,28)(26,34,29,39)(27,37,38,35)(31,36,40,33);(1,4,5,2)(3,7,8,6)(9,14,15,11)(10,16,17,24)(12,22,19,20)(13,23,18,21)(25,31,32,40)(26,35,29,37)(27,34,38,39)(28,36,30,33)

Polycyclic presentation:

2^9|b^2=h,c^2=gi,d^2=gh,e^2=gh,f^2=h,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, 9632030);