Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32932 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1314 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(5,6)(7,8)(9,10,12,11)(14,18)(15,21)(16,23)(19,26)(20,25)(22,27);(9,11,12,10)(13,14,16,18,17,20,23,25)(15,19,21,24,22,26,27,28);(1,2,3,4)(15,22)(19,26)(21,27)(24,28);(6,8)(13,15,16,21,17,22,23,27)(14,19,18,24,20,26,25,28)Polycyclic presentation:
2^9|a^2=h,b^2=eh,c^2=f,d^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,g^a=gi,d^c=di104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 300379);