#8899285

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

4946 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=gh,e^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8806481);