Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #93537 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1294 |
| commutator | C22×C8 | C42 |
| Frattini, cosocle | C2×C4×M16 | C22 |
| prime radical | C24 | #106 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C2×C4×M16 ▷ C22×C4 ▷ C2 ▷ C1 | |
253 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 253 conjugacy classes of subgroups — too many to draw. Only its 68 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 — 2 generators on 24 points:
(1,2,5,3,4,7,6,8)(10,15)(11,19)(12,13)(16,20)(17,24)(18,23)(21,22);(1,3,6,7,4,8,5,2)(9,10,12,16,14,18,13,17)(11,20,21,23,22,24,19,15)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fhi,d^2=h,e^2=hi,f^2=i,g^2=i,b^a=bc,c^a=cf,e^a=efghi,f^a=fi,g^a=gi,c^b=cg,d^b=dh,f^b=fi,e^c=ei56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 828);