Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #153117 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2196 |
| commutator, Frattini, cosocle | C2×#127 | C23 |
| prime radical | C2×D42 | C22 |
| LCS | G ▷ C2×#127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ C2×#127 ◁ G | |
| LECS | G ▷ C2×#127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
613 conjugacy classes of subgroups; only the 48 normal subgroups are shown.
This group has 613 conjugacy classes of subgroups — too many to draw. Only its 48 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) = 32 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,23)(20,24);(1,3,4,8)(2,5,6,7)(9,11)(10,14)(12,17)(13,19)(15,21)(16,23)(18,20)(22,24);(12,20)(14,21)(22,23)Polycyclic presentation:
2^9|a^2=d,b^2=d,g^2=i,b^a=bd,c^a=ce,f^a=fghi,h^a=hi,c^b=cf,e^b=eg,g^b=gi,h^b=hi,g^c=gh,f^e=fhi,g^e=gi,g^f=gi41 conjugacy classes — view the full character table.
| maximal subgroup of | #72279654 |
| maximal normal subgroup of | #72279654 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60415);