Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417589 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1234 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(5,6)(7,8)(10,11)(13,14)(15,18)(16,19)(17,20)(21,25)(22,28)(23,27)(24,26);(9,10)(11,12)(13,15,21,22,16,23,17,24)(14,18,25,28,19,27,20,26);(1,2,3,4)(14,19)(15,22)(17,21)(18,26)(23,24)(27,28);(6,8)(9,11,12,10)(15,23)(18,27)(22,24)(26,28)Polycyclic presentation:
2^9|b^2=gi,c^2=fi,d^2=e,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=dei,g^c=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 345356);