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×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1654 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)(5,6,10,13)(7,14,16,11)(8,17,9,12)(15,19,18,20)(21,22,26,31)(23,28,34,43)(24,35,37,29)(25,30,40,47)(27,32,41,48)(33,49,51,42)(36,45,38,44)(39,52,50,46);(5,7,10,16)(6,11,13,14)(8,15,9,18)(12,19,17,20)(21,23,26,34)(22,28,31,43)(24,36,37,38)(25,27,40,41)(29,44,35,45)(30,32,47,48)(33,50,51,39)(42,52,49,46);(2,4)(21,24,27,39,26,37,41,50)(22,29,32,46,31,35,48,52)(23,33,25,36,34,51,40,38)(28,42,30,44,43,49,47,45);(5,8,10,9)(6,12,13,17)(7,15,16,18)(11,19,14,20)(24,37)(29,35)(33,51)(36,38)(39,50)(42,49)(44,45)(46,52);(5,9,10,8)(6,12,13,17)(7,15,16,18)(11,20,14,19)(21,25,27,34,26,40,41,23)(22,30,32,43,31,47,48,28)(24,38,39,33,37,36,50,51)(29,45,46,42,35,44,52,49)Polycyclic presentation:
2^9|a^2=fi,b^2=fi,c^2=h,d^2=f,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=df,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, 6718970);