Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806623 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #42475 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1835 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,14)(13,16)(18,22)(23,28)(25,29)(30,32);(9,11)(10,13)(12,16)(14,15)(18,23)(19,26)(21,24)(22,28)(25,29)(27,31);(2,5)(6,8)(10,14)(11,16)(17,18,24,28,20,22,21,23)(19,25,31,32,27,29,26,30);(10,14)(13,15)(18,22)(23,28)(25,29)(30,32);(1,3,4,7)(2,6,5,8)(17,19,24,31,20,27,21,26)(18,25,28,32,22,29,23,30)Polycyclic presentation:
2^9|c^2=fhi,e^2=gh,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di74 conjugacy classes — view the full character table.
| maximal subgroup of | #72313602, #72313671, #72313981 |
| maximal normal subgroup of | #72313602, #72313671, #72313981 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6737447);