Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9938 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #392 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C22 | #1438 |
| prime radical | C22×#41 | C2×C4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
1005 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,9,6,10,4,8)(3,11,13,7,14,15,12,16)(18,22)(20,27)(23,31)(24,28)(25,29)(30,32);(1,3)(2,7)(4,12)(5,13)(6,14)(8,11)(9,15)(10,16)(17,18)(19,23)(20,28)(21,24)(22,27)(25,30)(26,32)(29,31);(1,4,6,5)(2,8,10,9)(3,12,14,13)(7,15,16,11)(17,19,21,26)(18,23,24,32)(20,29,27,25)(22,30,28,31)Polycyclic presentation:
2^9|a^2=g,c^2=h,d^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,f^a=fhi,c^b=cf,d^b=dh,e^b=eh,d^c=di68 conjugacy classes — view the full character table.
| maximal subgroup of | #72316773, #72316783, #72338221, … |
| maximal normal subgroup of | #72316773, #72316783, #72338221, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 22399);