Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417558 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #35527 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2400 conjugacy classes of subgroups; only the 210 normal subgroups are shown.
This group has 2400 conjugacy classes of subgroups — too many to draw. Only its 210 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:
(1,2,4,6)(3,7,8,5)(9,10)(11,17,19,24)(12,15)(13,16)(14,18,22,21)(20,23);(1,3)(2,5)(4,8)(6,7)(12,20)(14,22)(15,23)(18,21);(9,11)(10,14)(12,21)(13,19)(15,24)(16,22)(17,23)(18,20);(3,8)(5,7)(12,20)(15,23)(17,24)(18,21)Polycyclic presentation:
2^9|a^2=eh,f^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,f^b=fi,f^c=fi,h^c=hi,f^d=fi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72233137, #72236058, #72256727, … |
| maximal normal subgroup of | #72233137, #72236058, #72256727, … |
| maximal quotient of | #72233126, #72233847 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 356723);