Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6499874 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #35346 |
| commutator, Frattini, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
4580 conjugacy classes of subgroups; only the 576 normal subgroups are shown.
This group has 4580 conjugacy classes of subgroups — too many to draw. Only its 576 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 5 generators on 16 points:
(1,2)(3,6)(4,7)(5,8)(9,13)(10,12)(11,15)(14,16);(6,12)(13,14);(1,3)(2,6)(4,9)(5,10)(7,14)(8,12)(11,16)(13,15);(1,4)(2,7)(3,9)(5,11)(6,13)(8,15)(10,16)(12,14);(1,3)(2,6)(4,9)(5,10)(7,13)(8,12)(11,16)(14,15)Polycyclic presentation:
2^9|b^a=bf,c^a=cg,h^a=hi,c^b=ch,e^b=eh,d^c=dh,f^c=fi,g^d=gi,f^e=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6407070);