Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10530 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #394 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C22 | #1442 |
| prime radical | C24 | #124 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
504 conjugacy classes of subgroups; only the 126 normal subgroups are shown.
This group has 504 conjugacy classes of subgroups — too many to draw. Only its 126 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:
(2,7,8,3)(10,14,16,20)(15,22,17,23);(1,2,5,7,6,8,4,3)(9,10)(11,17)(12,20)(13,16)(14,21)(15,19)(18,22)(23,24);(1,3,4,2,6,7,5,8)(9,11,12,18,13,19,21,24)(10,15,14,22,16,17,20,23)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=d,d^2=h,e^2=i,b^a=bd,c^a=ce,c^b=cf,d^b=dh,f^b=fi,g^b=gh,e^c=ei,f^c=fh,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72210001 |
| maximal normal subgroup of | #72210001 |
| maximal quotient of | #72239971, #72240026, #72240043, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 27895);