#8993582

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

4778 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8900778);