Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3055 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C2×C4 | C23×C8 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C22×C22⋊C4 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1474 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(5,6,8,11,9,7,10,12)(13,14)(15,22)(16,21)(17,26)(18,20)(19,23)(24,28)(25,27);(5,7,8,12,9,6,10,11)(13,15,17,23,18,24,16,25)(14,19,21,22,20,27,26,28);(1,2,3,4)(14,20)(19,27)(21,26)(22,28);(13,16,18,17)(14,21,20,26)(15,23,24,25)(19,28,27,22)Polycyclic presentation:
2^9|a^2=g,b^2=eg,c^2=fh,d^2=h,e^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,d^b=dh128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 111236);