Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8760 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #381 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#381 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1005 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,5,8,6,3,4,7)(9,10,15,19,13,20,14,18)(11,16,21,17,12,22,23,24);(9,11,13,12)(10,16,20,22)(14,21,15,23)(17,19,24,18);(1,3)(2,6)(4,8)(5,7)(9,12,13,11)(10,17)(14,23,15,21)(16,19)(18,22)(20,24);(1,4,6,5)(2,7,3,8)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=e,d^2=i,f^2=hi,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^b=gh,f^c=fi,g^c=gh86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 97553);