Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30396 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C4×#95 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1466 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,3)(9,10,11,12)(13,14,17,21,18,15,22,20)(16,19,24,27,25,23,28,26);(5,6)(7,8)(13,15)(14,18)(16,19)(17,21)(20,22)(23,25)(24,26)(27,28);(1,2,4,3)(6,8)(13,16,18,25)(14,19,15,23)(17,24,22,28)(20,26,21,27);(1,3,4,2)(14,20)(15,21)(17,22)(19,26)(23,27)(24,28)Polycyclic presentation:
2^9|a^2=h,c^2=fi,d^2=f,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,e^b=ei,h^b=hi,e^d=ei,h^d=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 292371);