Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #142256 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1595 |
| commutator, Frattini, cosocle | C22×M16 | C23 |
| LCS | G ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×M16 ◁ G | |
| LECS | G ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
724 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 724 conjugacy classes of subgroups — too many to draw. Only its 72 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:
(1,2)(3,4)(6,12)(7,17)(8,18)(9,10)(13,16)(14,19)(15,20);(5,6)(7,14)(8,13)(9,19)(10,15)(11,16)(12,17)(18,20);(2,4)(6,13)(7,8)(9,10)(12,16)(14,15)(17,18)(19,20)Polycyclic presentation:
2^9|d^2=gi,f^2=gi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,f^a=fh,g^a=gi,h^a=hi,c^b=cf,d^b=dg,f^b=fg,g^b=gi,d^c=dh,f^c=fg,g^c=gi,h^c=hi,f^d=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72250264, #72250587, #72250602 |
| maximal normal subgroup of | #72250264, #72250587, #72250602 |
| maximal quotient of | #72307697 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 50000);