#8968523

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

4680 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,e^2=g,f^2=g,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, 8875719);