#9786691

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

4724 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)(19,21)(20,25)(22,27)(23,28)(24,26)(29,30)(31,32);(1,2,4,6)(3,5,8,7)(9,10)(11,14)(12,15)(13,16)(17,19,23,26)(18,24,28,21)(20,27,30,31)(22,25,32,29);(1,2)(3,5)(4,6)(7,8)(9,11)(10,13)(12,16)(14,15)(17,20,23,30)(18,25,28,29)(19,27,26,31)(21,22,24,32);(9,10)(11,14)(12,15)(13,16)(17,21,23,24)(18,26,28,19)(20,22,30,32)(25,31,29,27);(1,2)(3,7)(4,6)(5,8)(9,10)(11,13)(12,15)(14,16)(17,22)(18,27)(19,29)(20,24)(21,30)(23,32)(25,26)(28,31);(1,3)(2,5)(4,8)(6,7)(9,11)(10,14)(12,16)(13,15)(17,18)(19,24)(20,25)(21,26)(22,31)(23,28)(27,32)(29,30)

Polycyclic presentation:

2^9|b^2=ghi,c^2=g,d^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, 9693887);