Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30441 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2668 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14407 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1093 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,10,8,11,6,26)(3,12,15,9,17,29,16,30)(4,18,20,5,22,24,21,25)(13,14,19,23,28,27,31,32);(1,3)(2,9)(4,19)(5,23)(6,16)(7,15)(8,17)(10,29)(11,30)(12,26)(13,21)(14,18)(20,28)(22,31)(24,27)(25,32);(1,4,8,22)(2,5,11,25)(3,13)(6,21,7,20)(9,27)(10,24,26,18)(12,23)(14,30)(15,19)(16,31)(17,28)(29,32);(1,5,8,25)(2,4,11,22)(3,14)(6,18,7,24)(9,28)(10,20,26,21)(12,19)(13,30)(15,23)(16,32)(17,27)(29,31)Polycyclic presentation:
2^9|a^2=h,c^2=ef,d^2=ef,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,g^b=gi,e^c=ei,f^c=fi,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 292733);