Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #135479 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1819 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C22×C4⋊C4 | C23 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×#123 ◁ G | |
| LECS | G ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
775 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 775 conjugacy classes of subgroups — too many to draw. Only its 70 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:
(1,2,5,8,4,6,3,7)(9,10)(11,18,20,14)(12,17)(13,16)(15,19,22,24)(21,23);(2,6)(7,8)(9,11)(10,14)(12,19)(13,20)(15,17)(16,18)(21,24)(22,23);(1,3,4,5)(2,7)(6,8)(9,11)(10,15,16,22)(12,19)(13,20)(14,17,18,23)(21,24)Polycyclic presentation:
2^9|a^2=f,c^2=dh,e^2=g,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,f^b=fg,e^c=ei,f^c=fgh,f^e=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72226006, #72226010, #72226011, … |
| maximal normal subgroup of | #72226006, #72226010, #72226011, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 42693);