#10143813

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

4966 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,d^2=g,e^2=i,f^2=g,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, 10051009);