Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #120728 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1442 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
695 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,5)(6,7)(9,10)(11,17,19,14)(12,20)(13,16)(15,22)(18,23,21,24);(1,2)(3,5)(4,6)(7,8)(9,11)(10,14)(12,21,22,18)(13,19)(15,24,20,23)(16,17);(2,6)(4,8)(5,7)(9,12,13,22)(10,15)(11,18,19,21)(14,23)(16,20)(17,24)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=di,f^2=hi,b^a=bd,c^a=ce,d^a=dh,c^b=cf,f^b=fi,g^b=gh,e^c=eh,f^c=fhi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 27925);