Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #196038 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C25 | C24 |
| prime radical | C22×#120 | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C22⋊C4 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
2500 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,3)(5,6)(7,8)(9,10)(11,17)(12,20,22,15)(13,16)(14,19)(18,24,21,23);(1,2,4,3)(9,11,13,19)(10,14,16,17)(12,21)(15,23)(18,22)(20,24);(6,8)(9,12)(10,15)(11,18)(13,22)(14,23)(16,20)(17,24)(19,21);(1,3,4,2)(10,16)(14,17)(15,20)(23,24)Polycyclic presentation:
2^9|a^2=h,b^2=eh,d^2=ei,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gi,h^c=hi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72313943, #72313958, #72313985 |
| maximal normal subgroup of | #72313943, #72313958, #72313985 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 272172);