Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #484194 |
| 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 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×C4⋊C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
1043 conjugacy classes of subgroups; only the 144 normal subgroups are shown.
This group has 1043 conjugacy classes of subgroups — too many to draw. Only its 144 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)(10,15)(11,13)(12,19)(16,24)(17,23)(18,21)(20,22);(1,2)(3,5)(4,6)(7,8)(9,10)(11,16)(12,20)(13,22)(14,17)(15,19)(18,23)(21,24);(1,3,4,7)(2,5,6,8);(9,11,14,13)(10,16,17,22)(12,18,21,19)(15,20,23,24)Polycyclic presentation:
2^9|c^2=h,d^2=i,e^2=gi,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eg,g^a=gi,e^b=eg,f^b=fh,g^b=gi,e^c=eh56 conjugacy classes — view the full character table.
| maximal subgroup of | #72280614, #72280634, #72280653 |
| maximal normal subgroup of | #72280614, #72280634, #72280653 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 391818);