#9787160

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

4897 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=g,d^2=h,e^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, 9694356);