Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #213564 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C25 | C24 |
| prime radical | C22×#120 | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C22⋊C4 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
2157 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6)(7,9)(8,11)(10,12)(13,14)(15,21,23,18)(16,24,25,19)(17,20)(22,26)(27,28);(1,2)(3,4)(5,7)(6,8)(9,12)(10,11);(2,4)(5,8,10,9)(6,11,12,7)(13,15)(14,18)(16,22)(17,23)(19,26)(20,21)(24,27)(25,28);(5,9,10,8)(6,11,12,7)(13,16,17,25)(14,19,20,24)(15,22,23,28)(18,26,21,27)Polycyclic presentation:
2^9|a^2=h,c^2=e,d^2=ei,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=de,h^c=hi,h^d=hi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72313964, #72313971, #72313985 |
| maximal normal subgroup of | #72313964, #72313971, #72313985 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 290890);