Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806839 |
| 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 | |
1682 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)(24,30,41,50)(25,42,28,31)(26,32,39,52)(35,51,43,45)(36,46,44,48)(37,49,40,47);(5,7)(6,10)(8,14)(9,15)(11,18)(12,13)(16,19)(17,20)(21,23,27,38)(22,29,33,34)(24,39,41,26)(25,36,28,44)(30,32,50,52)(31,46,42,48)(35,40,43,37)(45,47,51,49);(2,4)(21,24,25,40,27,41,28,37)(22,30,42,47,33,50,31,49)(23,35,44,26,38,43,36,39)(29,45,46,52,34,51,48,32);(5,8,9,17)(6,11,12,16)(7,14,15,20)(10,18,13,19)(21,25,27,28)(22,31,33,42)(23,36,38,44)(24,40,41,37)(26,35,39,43)(29,46,34,48)(30,49,50,47)(32,45,52,51);(7,15)(10,13)(14,20)(18,19)(21,26,28,35,27,39,25,43)(22,32,31,51,33,52,42,45)(23,37,36,41,38,40,44,24)(29,47,48,30,34,49,46,50)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=hi,d^2=f,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6716023);