Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #490428 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#459 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
646 conjugacy classes of subgroups; only the 154 normal subgroups are shown.
This group has 646 conjugacy classes of subgroups — too many to draw. Only its 154 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:
(2,5)(3,6)(7,8)(10,15)(11,13)(12,19)(16,24)(17,23)(18,21)(20,22);(1,2,6,8,4,7,3,5);(9,10)(11,16)(12,20)(13,22)(14,17)(15,19)(18,23)(21,24);(9,11,14,13)(10,16,17,22)(12,18,21,19)(15,20,23,24)Polycyclic presentation:
2^9|b^2=eg,d^2=i,e^2=g,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,e^a=eg,f^a=fh,h^a=hi,f^c=fh,h^c=hi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72216361, #72216593, #72225213, … |
| maximal normal subgroup of | #72216361, #72216593, #72225213, … |
| maximal quotient of | #72216276, #72226933 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 397651);