#9788898

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

4155 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=gh,d^2=g,e^2=g,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, 9696094);