Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C23×D4 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2274 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(5,6)(7,10)(8,12)(9,11)(13,14)(15,22)(16,20)(17,26)(18,21)(19,24)(23,25)(27,28);(5,7,9,8)(6,10,11,12)(13,15)(14,19)(16,24)(17,27)(18,25)(20,23)(21,28)(22,26);(5,8,9,7)(6,10,11,12);(13,14)(15,23)(16,20)(17,26)(18,21)(19,27)(22,25)(24,28);(13,16,18,17)(14,20,21,26)(15,24,25,27)(19,23,28,22)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=g,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,d^b=df,f^b=fi,f^d=fi116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6287673);