Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8304 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×C22⋊C4 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C22×C22⋊C4 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1370 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)(5,6,9,10,8,11,7,12)(13,14)(15,21,23,18)(16,19)(17,20)(22,27,28,25)(24,26);(1,2,3,4)(5,7,8,9)(6,10,11,12);(6,11)(7,9)(13,15)(14,18)(16,22)(17,23)(19,25)(20,21)(24,28)(26,27);(13,16,17,24)(14,19,20,26)(15,22,23,28)(18,25,21,27)Polycyclic presentation:
2^9|a^2=g,b^2=e,d^2=hi,g^2=i,b^a=be,c^a=cf,f^a=fh,c^b=ci,g^c=gh104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 95480);