Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #213564 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C22×C4 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1216 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6)(7,10)(8,11)(9,12)(13,14,18,19,17,21,15,22)(16,24,23,27,25,20,26,28);(1,2)(3,4)(5,7,10,6)(8,12,9,11)(13,15,17,18)(14,19,21,22)(16,23,25,26)(20,27,24,28);(2,4)(5,8,10,9)(6,11,7,12)(13,16,17,25)(14,20,21,24)(15,23,18,26)(19,27,22,28);(5,9,10,8)(6,11,7,12)(15,18)(19,22)(23,26)(27,28)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=e,d^2=ei,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=de,h^c=hi,h^d=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 291016);