Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6628896 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
17912 conjugacy classes of subgroups; only the 946 normal subgroups are shown.
This group has 17912 conjugacy classes of subgroups — too many to draw. Only its 946 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2)(3,4)(10,16)(17,20);(5,6)(7,10)(8,11)(9,12)(13,17)(14,16)(15,18)(19,20);(5,7)(6,10)(8,13)(9,14)(11,17)(12,16)(15,19)(18,20);(2,4)(5,8)(6,11)(7,13)(9,15)(10,17)(12,18)(14,19)(16,20);(8,15)(11,18)(13,19)(17,20)Polycyclic presentation:
2^9|b^a=bf,c^a=cg,d^a=dh,g^b=gi,f^c=fi,e^d=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6536092);