Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #135346 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1824 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C24 | C2×SD16 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
373 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 373 conjugacy classes of subgroups — too many to draw. Only its 88 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:
(3,6,7,8)(10,15)(11,19)(12,13)(16,23)(17,21)(18,24)(20,22);(1,2,5,4)(3,6,7,8)(11,20)(15,23)(19,22)(21,24);(1,3)(2,6)(4,8)(5,7)(9,10,12,18,14,16,13,17)(11,21,19,23,20,24,22,15)Polycyclic presentation:
2^9|a^2=f,b^2=gh,c^2=dg,e^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eg,g^a=gi,h^a=hi,e^b=ei,d^c=di,e^c=eh,f^c=fgh,g^c=gi,h^c=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72240841, #72265403, #72265406, … |
| maximal normal subgroup of | #72240841, #72265403, #72265406, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 42588);