#10149916

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

4686 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:

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

Polycyclic presentation:

2^9|b^2=ghi,c^2=g,d^2=g,e^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

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, 10057112);