Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #486291 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#447 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #24267 | C2 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×C8×D4 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
1001 conjugacy classes of subgroups; only the 150 normal subgroups are shown.
This group has 1001 conjugacy classes of subgroups — too many to draw. Only its 150 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) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(5,8)(10,14)(11,17)(12,19)(15,24)(16,23)(18,21)(20,22);(1,2)(3,5)(4,6)(7,8)(9,10,13,16)(11,18,20,24)(12,21,19,15)(14,22,23,17);(1,3)(2,5)(4,7)(6,8);(9,11,19,22,13,20,12,17)(10,14,21,24,16,23,15,18)Polycyclic presentation:
2^9|b^2=i,d^2=gi,e^2=gi,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eg,g^a=gi,d^b=dg,e^b=eg,f^b=fh,g^b=gi,e^c=eh65 conjugacy classes — view the full character table.
| maximal subgroup of | #72280653 |
| maximal normal subgroup of | #72280653 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 393491);