#8990898

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

4610 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=i,d^2=gi,e^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, 8898094);