Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476092 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#127 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1172 conjugacy classes of subgroups; only the 194 normal subgroups are shown.
This group has 1172 conjugacy classes of subgroups — too many to draw. Only its 194 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:
(5,6,8,7)(9,10)(11,14)(12,21)(13,17)(15,18)(16,19)(20,23)(22,24);(1,2)(3,4)(5,6)(7,8)(9,11,15,19,16,18,14,10)(12,17,13,21,23,24,22,20);(2,4)(5,7,8,6)(9,12)(10,17)(11,20)(13,14)(15,22)(16,23)(18,21)(19,24);(6,7)(9,13,15,23,16,22,14,12)(10,17,11,21,19,24,18,20)Polycyclic presentation:
2^9|a^2=ef,b^2=f,c^2=ef,d^2=f,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=def,d^c=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 385388);