#9781211

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

4516 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=gh,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, 9688407);