Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12173 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1598 |
| commutator | C2×M16 | C22×C4 |
| Frattini, cosocle | C22×M16 | C23 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×M16 ◁ G | |
| LECS | G ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
568 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 568 conjugacy classes of subgroups — too many to draw. Only its 86 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) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(2,4)(5,6,7,13,8,14,18,20,10,15,17,19,9,12,16,11);(1,2,3,4)(6,11)(7,16)(8,9)(12,13)(14,19)(15,20)(17,18);(6,12,15,14)(11,13,20,19)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,c^2=h,d^2=gi,e^2=i,f^2=gi,g^2=i,b^a=bd,c^a=ce,e^a=ehi,h^a=hi,d^b=dg,e^b=ehi,f^b=fg,g^b=gi,d^c=dh,f^c=fh,e^d=ei,f^e=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72313480, #72313487, #72313551 |
| maximal normal subgroup of | #72313480, #72313487, #72313551 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44396);