#9786721

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

5226 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|b^2=hi,c^2=g,d^2=g,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, 9693917);