Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3026 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator | C23 | C23×C8 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C4×#120 |
| prime radical | C2×B(4,𝔽2) | C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1216 conjugacy classes of subgroups; only the 268 normal subgroups are shown.
This group has 1216 conjugacy classes of subgroups — too many to draw. Only its 268 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:
(1,2,4,7,5,3,6,8)(9,10,13,16)(11,14,18,21)(12,15,19,22)(17,23,24,20);(9,11)(10,14)(12,17)(13,18)(15,20)(16,21)(19,24)(22,23);(9,12)(10,15)(11,17)(13,19)(14,20)(16,22)(18,24)(21,23);(1,3,4,8,5,2,6,7)(10,16)(14,21)(15,22)(20,23)Polycyclic presentation:
2^9|a^2=g,d^2=gh,g^2=i,b^a=be,c^a=cf,d^a=dh,f^b=fh,e^c=eh116 conjugacy classes — view the full character table.
| maximal subgroup of | #72315965, #72315968, #72315990, … |
| maximal normal subgroup of | #72315965, #72315968, #72315990, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 116005);