Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | #34963 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4874 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(9,10,13,14)(11,12,15,16)(17,18)(19,22)(20,24)(21,29)(23,26)(25,31)(27,32)(28,30);(1,2,5,4)(3,7,8,6)(10,14)(11,15)(17,18)(19,22)(20,24)(21,25)(23,26)(27,30)(28,32)(29,31);(2,4)(6,7)(9,11)(10,12)(13,15)(14,16)(17,19,23,20)(18,24,26,22)(21,27,31,28)(25,32,29,30);(1,3)(2,6)(4,7)(5,8)(9,12)(10,11)(13,16)(14,15)(17,20,23,19)(18,24,26,22)(21,28,31,27)(25,32,29,30);(17,21,23,31)(18,25,26,29)(19,27,20,28)(22,30,24,32);(1,4,5,2)(3,7,8,6)(17,22,23,24)(18,19,26,20)(21,30,31,32)(25,27,29,28)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=h,d^2=h,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8103024);