Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #186146 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#379 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#3499 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
3207 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,6,5,7,3,8)(9,10,14,17,15,12,13,22)(11,18,20,16,21,24,23,19);(2,6)(3,4)(7,8)(9,11,15,21)(10,16,12,19)(13,23,14,20)(17,24,22,18);(9,12)(10,15)(11,19)(13,22)(14,17)(16,21)(18,23)(20,24);(2,7)(6,8)Polycyclic presentation:
2^9|a^2=g,b^2=h,e^2=i,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dhi,e^a=eh,e^b=ei,g^b=ghi,f^c=fh,g^c=gh74 conjugacy classes — view the full character table.
| maximal subgroup of | #72338220, #72338222, #72338224 |
| maximal normal subgroup of | #72338220, #72338222, #72338224 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 93500);