Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #107334 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22⋊#95 |
| commutator | C42 | C2×C42 |
| Frattini, cosocle | C22×C42 | C23 |
| prime radical | C22×#127 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
1180 conjugacy classes of subgroups; only the 190 normal subgroups are shown.
This group has 1180 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) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,5,6,7)(9,10,13,16,14,18,12,17)(11,19,21,23,22,24,20,15);(1,2)(3,7)(4,6)(5,8)(9,11)(10,15,18,23)(12,20)(13,21)(14,22)(16,19,17,24);(3,8)(5,7)(10,16)(12,13)(15,19)(17,18)(20,21)(23,24)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=hi,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dh,g^a=gh,d^b=di,e^b=ehi,f^b=fi,d^c=dhi,e^c=eh,f^c=fh68 conjugacy classes — view the full character table.
| maximal subgroup of | #72258038, #72258064, #72286226, … |
| maximal normal subgroup of | #72258038, #72258064, #72286226, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 14535);