Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #196038 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27×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 | C2×#2404 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C22⋊C4 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
3234 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(5,6)(7,8)(9,10)(11,14)(12,17,20,15)(13,18)(16,21)(19,22,24,23);(2,4)(9,11,14,10)(12,15)(13,18,21,16)(17,20)(19,22)(23,24);(6,8)(9,12)(10,15)(11,17)(13,19)(14,20)(16,22)(18,23)(21,24);(2,4)(9,13,14,21)(10,16,11,18)(12,19,20,24)(15,22,17,23)Polycyclic presentation:
2^9|a^2=h,b^2=hi,d^2=i,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 | #72261691, #72262091, #72265217, … |
| maximal normal subgroup of | #72261691, #72262091, #72265217, … |
| maximal quotient of | #72265993 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 271670);