Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #482716 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#99 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C4⋊C4 | C24 |
| prime radical | #36091 | C2 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×Q8 ◁ C2×#550 ◁ G | |
| LECS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1722 conjugacy classes of subgroups; only the 198 normal subgroups are shown.
This group has 1722 conjugacy classes of subgroups — too many to draw. Only its 198 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:
(1,2,4,8,5,7,3,6)(9,10)(11,20)(12,17)(13,18)(14,23)(15,19)(16,21)(22,24);(1,3,5,4)(2,6)(7,8)(9,11,15,12)(10,16,19,24)(13,21,14,22)(17,23,20,18);(9,12,15,11)(10,17,19,20)(13,21,14,22)(16,23,24,18);(9,13,15,14)(10,18,19,23)(11,21,12,22)(16,17,24,20)Polycyclic presentation:
2^9|a^2=f,b^2=gi,c^2=hi,d^2=hi,e^2=hi,f^2=i,b^a=be,e^a=eg,g^a=gi,e^b=eh,f^b=fghi,d^c=dhi,f^e=fi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 390139);