Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | #51104 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4687 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2)(3,6)(4,5)(7,8)(9,10,13,15)(11,14,12,16);(2,5)(6,8)(17,18,21,24)(19,25,27,22)(20,28,30,23)(26,31,29,32);(9,10)(11,16)(12,14)(13,15)(17,19,21,27)(18,22,24,25)(20,29,30,26)(23,32,28,31);(9,11,13,12)(10,14,15,16)(17,20,21,30)(18,23,24,28)(19,26,27,29)(22,31,25,32);(9,12,13,11)(10,14,15,16)(19,27)(22,25)(26,29)(31,32);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=i,d^2=hi,e^2=h,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^b=di,d^c=dhi,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7614554);