Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #246415 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1548 conjugacy classes of subgroups; only the 190 normal subgroups are shown.
This group has 1548 conjugacy classes of subgroups — too many to draw. Only its 190 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,13)(12,15)(14,16);(1,3,4,8)(2,5,6,7)(11,16)(13,14)(17,18)(19,21)(20,23)(22,24);(10,13)(11,16)(14,15)(18,21)(19,24)(22,23);(2,6)(5,7)Polycyclic presentation:
2^9|b^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,c^b=cg,f^b=fh,g^b=gi,f^c=fh,g^c=gi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72264304, #72264308, #72264352 |
| maximal normal subgroup of | #72264304, #72264308, #72264352 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 158691);