Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23848 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×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 | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1610 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)(5,6,7,8)(9,10,12,11)(15,22)(19,27)(23,28)(24,25);(2,4)(9,11,12,10)(13,14)(15,23)(16,20)(17,21)(18,26)(19,25)(22,28)(24,27);(6,8)(13,15,18,25,17,22,16,24)(14,19,20,23,21,27,26,28);(2,4)(13,16,17,18)(14,20,21,26)(15,24,22,25)(19,23,27,28)Polycyclic presentation:
2^9|a^2=h,b^2=fhi,c^2=g,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 275634);