Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806623 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1693 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)(5,6,9,12)(7,13,15,10)(8,16,17,11)(14,18,20,19)(21,22,27,33)(23,34,38,29)(24,30,40,51)(25,41,28,31)(26,32,42,49)(35,52,43,45)(36,46,44,48)(37,50,39,47);(5,7)(6,10)(8,14)(9,15)(11,18)(12,13)(16,19)(17,20)(21,23,27,38)(22,29,33,34)(24,26,40,42)(25,36,28,44)(30,49,51,32)(31,46,41,48)(35,37,43,39)(45,50,52,47);(2,4)(21,24,25,37,27,40,28,39)(22,30,41,50,33,51,31,47)(23,35,44,42,38,43,36,26)(29,45,46,32,34,52,48,49);(5,8,9,17)(6,11,12,16)(7,14,15,20)(10,18,13,19)(21,25,27,28)(22,31,33,41)(23,36,38,44)(24,39,40,37)(26,35,42,43)(29,46,34,48)(30,50,51,47)(32,45,49,52);(7,15)(10,13)(14,20)(18,19)(21,26,25,35,27,42,28,43)(22,32,41,52,33,49,31,45)(23,37,44,40,38,39,36,24)(29,47,46,30,34,50,48,51)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=hi,d^2=f,e^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6715776);