#9062543

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

4784 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=gh,d^2=i,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8969739);