Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32932 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1294 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(5,6)(7,8)(9,10,12,11)(14,20)(15,16)(17,18)(21,28)(23,24)(25,26);(9,11,12,10)(13,14,17,20,19,24,18,23)(15,21,16,27,26,28,25,22);(1,2,3,4)(13,15,18,25,19,26,17,16)(14,21,23,22,24,28,20,27);(6,8)(13,16,17,26,19,25,18,15)(14,22,20,21,24,27,23,28)Polycyclic presentation:
2^9|a^2=h,b^2=eh,c^2=fi,d^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fi,g^a=gi,d^b=di104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 300412);