Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394329 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1576 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(5,6,9,11)(7,12,8,10)(13,14)(15,19)(16,25)(17,26)(18,20)(21,23)(22,27)(24,28);(5,7,9,8)(6,10,11,12)(13,15)(14,19)(16,21)(17,22)(18,24)(20,28)(23,25)(26,27);(5,8,9,7)(6,10,11,12)(13,16,25,14,18,26,17,20)(15,21,22,28,24,27,23,19);(2,4)(13,17,18,25)(14,16,20,26)(15,22,24,23)(19,21,28,27)Polycyclic presentation:
2^9|a^2=e,b^2=e,c^2=efi,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^c=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 305250);