#10216729

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

4712 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,d^2=i,e^2=gi,f^2=ghi,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, 10123925);