Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #496348 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #21794 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×#41 | C24 |
| prime radical | #60719 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #2875 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
762 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)(3,12)(4,9)(5,19,20,8)(6,11)(7,15)(10,17)(13,23)(14,25)(16,21,30,28)(18,24,32,26)(22,31,27,29);(1,3,4,15,6,14,17,13)(2,7,10,12,11,23,9,25)(5,16,18,31,20,30,32,29)(8,22,24,28,19,27,26,21);(2,8)(3,13)(4,17)(7,21)(9,24)(10,26)(11,19)(12,27)(14,15)(16,29)(18,32)(22,25)(23,28)(30,31);(2,9,11,10)(3,14)(5,20)(7,12,23,25)(8,24,19,26)(13,15)(18,32)(21,27,28,22)Polycyclic presentation:
2^9|a^2=g,b^2=e,d^2=h,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,h^a=hi,c^b=ce,d^b=di,e^c=ei,g^c=gh,f^d=fi,g^f=gi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 403544);