Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393909 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1618 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,3,4)(6,7)(13,14,17,21,16,18,25,19)(15,22,24,27,23,28,26,20);(5,6,8,7)(9,10)(11,12)(14,18)(19,21)(20,27)(22,28);(2,4)(5,7,8,6)(10,12)(14,19)(17,25)(18,21)(20,22)(24,26)(27,28);(13,15,16,23)(14,20,18,27)(17,26,25,24)(19,22,21,28)Polycyclic presentation:
2^9|a^2=efi,b^2=ei,c^2=ei,d^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,f^c=fi,h^c=hi,f^d=fi,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 303165);