Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #129617 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1790 |
| commutator | C22×D4 | C22×C4 |
| Frattini, prime radical, cosocle | C22×#41 | C23 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
837 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 837 conjugacy classes of subgroups — too many to draw. Only its 138 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:
(9,10,12,16,13,15,19,23)(11,14,18,22,17,21,20,24);(1,2,4,6)(3,7,8,5)(10,14)(12,19)(15,21)(16,22)(18,20)(23,24);(1,3,4,8)(2,5,6,7)(10,15)(11,17)(12,19)(14,21)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=f,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,h^a=hi,c^b=cf,e^b=ehi,g^b=gh,d^c=dh,g^c=gi,e^d=ei,g^d=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72303570 |
| maximal normal subgroup of | #72303570 |
| maximal quotient of | #72303618, #72303621 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56678);