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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1632 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)(21,22,27,26)(23,28,33,41)(24,29,35,37)(25,30,40,39)(31,34,42,46)(32,38,44,51)(36,43,48,50)(45,47,49,52);(5,7,10,16)(6,11,13,14)(8,15,9,18)(12,19,17,20)(21,23,27,33)(22,28,26,41)(24,34,35,46)(25,38,40,51)(29,42,37,31)(30,44,39,32)(36,49,48,45)(43,52,50,47);(2,4)(21,24,26,37,27,35,22,29)(23,31,28,34,33,42,41,46)(25,36,39,50,40,48,30,43)(32,45,38,47,44,49,51,52);(5,8,10,9)(6,12,13,17)(7,15,16,18)(11,19,14,20)(24,35)(29,37)(31,42)(34,46)(36,48)(43,50)(45,49)(47,52);(5,9,10,8)(6,12,13,17)(7,15,16,18)(11,20,14,19)(21,25,22,30,27,40,26,39)(23,32,41,51,33,44,28,38)(24,36,29,43,35,48,37,50)(31,45,46,52,42,49,34,47)Polycyclic presentation:
2^9|a^2=fi,b^2=fi,c^2=h,d^2=f,e^2=fhi,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, 6718918);