#9779755

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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=g,d^2=g,e^2=gh,f^2=g,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, 9686951);