Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #383745 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14363 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60423 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1352 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,12,7,11,5,21)(3,13,18,29,16,8,17,25)(4,10,20,26,19,9,22,27)(14,30,15,24,31,23,32,28);(1,3)(2,8)(4,15)(5,18)(6,17)(7,16)(9,24)(10,28)(11,13)(12,29)(14,20)(19,32)(21,25)(22,31)(23,26)(27,30);(1,4)(2,9,11,10)(3,14)(5,22)(6,20)(7,19)(8,23,13,30)(12,27,21,26)(15,18)(16,31)(17,32)(24,25,28,29);(1,4)(2,10,11,9)(3,15)(5,20)(6,22)(7,19)(8,24,13,28)(12,27,21,26)(14,18)(16,32)(17,31)(23,25,30,29)Polycyclic presentation:
2^9|a^2=h,c^2=e,d^2=e,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,g^a=gi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,f^c=fi,g^c=gi,f^d=fi,h^d=hi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 290943);