Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806671 |
| 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 | |
1677 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)(23,34,37,29)(24,30,35,44)(25,39,28,31)(26,41,42,32)(36,45,51,49)(38,48,50,46)(40,47,43,52);(5,7)(6,10)(8,14)(9,15)(11,18)(12,13)(16,19)(17,20)(21,23,27,37)(22,29,33,34)(24,25,35,28)(26,38,42,50)(30,31,44,39)(32,48,41,46)(36,43,51,40)(45,47,49,52);(2,4)(21,24,28,23,27,35,25,37)(22,30,31,34,33,44,39,29)(26,38,43,36,42,50,40,51)(32,46,47,49,41,48,52,45);(5,8,9,17)(6,11,12,16)(7,14,15,20)(10,18,13,19)(21,25,27,28)(22,31,33,39)(23,35,37,24)(26,40,42,43)(29,44,34,30)(32,47,41,52)(36,50,51,38)(45,48,49,46);(7,15)(10,13)(14,20)(18,19)(21,26,25,40,27,42,28,43)(22,32,39,52,33,41,31,47)(23,36,24,38,37,51,35,50)(29,45,44,48,34,49,30,46)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=h,d^2=f,e^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6715792);