Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32700 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2760 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14375 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1056 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,10,6,9,19,24)(3,7,13,22,12,20,17,25)(4,14,16,26,15,8,18,23)(11,27,29,32,28,21,30,31);(7,20)(11,28)(22,25)(29,30);(1,3)(2,7,9,20)(4,11)(5,17)(6,12)(8,21,14,27)(10,25,24,22)(13,19)(15,28)(16,30)(18,29)(23,32,26,31);(1,4)(2,8)(3,11)(5,18)(6,15)(7,21)(9,14)(10,26)(12,28)(13,30)(16,19)(17,29)(20,27)(22,32)(23,24)(25,31)Polycyclic presentation:
2^9|a^2=h,c^2=ef,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^c=ei,f^c=fi,h^c=hi,g^d=gi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 291447);