Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | #34636 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1796 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(1,2)(3,4)(5,6)(7,13)(8,12)(9,11)(10,14)(15,19)(16,18)(17,20)(21,22,25,29,26,30,36,42)(23,27,33,39,32,38,45,49)(24,28,35,41,34,40,46,50)(31,37,44,48,43,47,51,52);(5,7)(6,10)(8,15)(9,14)(11,13)(12,17)(16,20)(18,19)(21,23,25,33,26,32,36,45)(22,27,29,39,30,38,42,49)(24,31,35,44,34,43,46,51)(28,37,41,48,40,47,50,52);(2,4)(7,14)(10,13)(15,20)(17,19)(21,24)(22,28)(23,31)(25,35)(26,34)(27,37)(29,41)(30,40)(32,43)(33,44)(36,46)(38,47)(39,48)(42,50)(45,51)(49,52);(6,11)(10,13)(12,18)(17,19)(24,34)(28,40)(31,43)(35,46)(37,47)(41,50)(44,51)(48,52);(5,8,9,16)(6,12,11,18)(7,15,14,20)(10,17,13,19)(23,32)(27,38)(31,43)(33,45)(37,47)(39,49)(44,51)(48,52)Polycyclic presentation:
2^9|a^2=h,b^2=h,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=cf,e^b=ei,d^c=di104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6283125);