Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23848 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C22×D8 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C4×C4⋊C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
748 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,4,3)(5,6,9,11,8,7,10,12)(13,14)(15,22)(16,20)(17,19)(18,23)(21,25)(24,26)(27,28);(2,3)(5,7,9,12,8,6,10,11)(13,15,16,23)(14,18,20,22)(17,24,25,28)(19,27,21,26);(6,7)(11,12)(14,19)(15,23)(17,25)(18,26)(20,21)(22,27);(1,3,4,2)(15,23)(18,22)(24,28)(26,27)Polycyclic presentation:
2^9|a^2=g,b^2=eg,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,c^b=chi,d^b=de,f^c=fh104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 136272);