Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806647 |
| 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×#469 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1709 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)(23,28)(24,29)(25,40)(26,31)(27,32)(30,34)(33,42)(35,43)(36,44)(37,45)(38,46)(39,48)(41,47)(49,51)(50,52);(5,7)(6,10)(8,14)(9,15)(11,18)(12,13)(16,19)(17,20)(21,23,27,35)(22,28,32,43)(24,36,38,41)(25,37,34,26)(29,44,46,47)(30,31,40,45)(33,49,50,39)(42,51,52,48);(2,4)(21,24,37,49,27,38,26,39)(22,29,45,51,32,46,31,48)(23,33,25,41,35,50,34,36)(28,42,40,47,43,52,30,44);(5,8,9,17)(6,11,12,16)(7,14,15,20)(10,18,13,19)(24,38)(29,46)(33,50)(36,41)(39,49)(42,52)(44,47)(48,51);(7,15)(10,13)(14,20)(18,19)(21,25,26,23,27,34,37,35)(22,30,31,43,32,40,45,28)(24,36,39,50,38,41,49,33)(29,47,48,42,46,44,51,52)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,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6715847);