Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #196038 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C42 |
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×#121 | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4⋊C4 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
2214 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,3)(6,8)(9,10)(11,14)(12,19,21,15)(13,16)(17,22,20,24)(18,23);(1,2,4,3)(9,11,13,18)(10,14,16,23)(12,20)(15,22)(17,21)(19,24);(5,6,7,8)(9,12)(10,15)(11,17)(13,21)(14,22)(16,19)(18,20)(23,24);(1,3,4,2)Polycyclic presentation:
2^9|a^2=h,b^2=ehi,c^2=fg,d^2=e,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 | #72313975, #72313993, #72314039 |
| maximal normal subgroup of | #72313975, #72313993, #72314039 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 276311);