#9788619

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

4340 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=g,d^2=g,e^2=g,f^2=h,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, 9695815);