Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #478457 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2760 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | C2×#534 | C22 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
1005 conjugacy classes of subgroups; only the 146 normal subgroups are shown.
This group has 1005 conjugacy classes of subgroups — too many to draw. Only its 146 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)(3,5,7,8)(9,10)(11,17,19,14)(12,15,21,24)(13,16)(18,23)(20,22);(1,3)(2,5)(4,7)(6,8)(9,11)(10,14)(12,20,21,18)(13,19)(15,23,24,22)(16,17);(2,6)(5,8)(9,12)(10,15,16,24)(11,18)(13,21)(14,22,17,23)(19,20);(3,7)(5,8)(9,11)(10,14)(12,18)(13,19)(15,23)(16,17)(20,21)(22,24)Polycyclic presentation:
2^9|a^2=efg,b^2=eh,c^2=e,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=def,g^b=gi,d^c=de,f^c=fi,h^c=hi,g^d=gi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72238059 |
| maximal normal subgroup of | #72238059 |
| maximal quotient of | #72237344 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 386947);