Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #133093 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1781 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C22×C4⋊C4 | C23 |
| prime radical | C2×#2430 | C2 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×#123 ◁ G | |
| LECS | G ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1643 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 1643 conjugacy classes of subgroups — too many to draw. Only its 86 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:
(1,2,4,8,5,7,3,6)(9,10)(11,18,20,15)(12,16)(13,17)(14,19,23,21)(22,24);(1,3,5,4)(2,6)(7,8)(9,11)(10,14)(12,21)(13,20)(15,16)(17,23)(18,24)(19,22);(9,11)(10,15)(12,19)(13,20)(14,16)(17,18)(21,22)(23,24)Polycyclic presentation:
2^9|a^2=f,b^2=gi,d^2=hi,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,g^a=gi,d^b=dh,f^b=fghi,f^c=fhi,f^d=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72226006, #72226010, #72226015, … |
| maximal normal subgroup of | #72226006, #72226010, #72226015, … |
| maximal quotient of | #72268846 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 40339);