Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #480872 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#99 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C22×Q8 | C24 |
| prime radical | C2×#550 | C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×Q8 ◁ C2×Q82 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
964 conjugacy classes of subgroups; only the 210 normal subgroups are shown.
This group has 964 conjugacy classes of subgroups — too many to draw. Only its 210 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:
(2,6)(7,8)(9,10,12,16,13,17,21,24)(11,15,19,23,18,14,20,22);(1,2,5,6)(3,7,4,8);(1,3,5,4)(2,7,6,8)(10,14)(11,18)(15,17)(16,23)(19,20)(22,24);(1,4,5,3)(2,7,6,8)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=e,d^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,d^b=de,f^c=fi,g^c=ghi,g^f=gi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72271403 |
| maximal normal subgroup of | #72271403 |
| maximal quotient of | #72271424 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 404850);