Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9303 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #386 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C2×#386 |
| prime radical | #3444 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1436 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,7,5,8,3,6)(9,10,14,18,15,12,13,17)(11,19,21,20,23,24,22,16);(2,6,8,7)(9,11,15,23)(10,16,12,20)(13,21,14,22)(17,19,18,24);(1,3,5,4)(2,6,8,7)(9,12)(10,15)(11,20)(13,18)(14,17)(16,23)(19,22)(21,24);(1,3,5,4)(2,6,8,7)(9,13,15,14)(10,17,12,18)(11,21,23,22)(16,19,20,24)Polycyclic presentation:
2^9|a^2=g,b^2=eg,c^2=h,d^2=hi,e^2=hi,f^2=i,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fi,e^b=ei,g^b=gi,f^c=fi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 106624);