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 | |
1652 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)(21,22,26,30)(23,27,33,40)(24,28,36,43)(25,29,37,44)(31,38,46,50)(32,39,35,42)(34,41,47,51)(45,49,48,52);(5,7,10,16)(6,11,13,14)(8,15,9,18)(12,19,17,20)(23,31)(24,34)(25,37)(27,38)(28,41)(29,44)(32,35)(33,46)(36,47)(39,42)(40,50)(43,51);(2,4)(21,23,25,31,26,33,37,46)(22,27,29,38,30,40,44,50)(24,35,34,48,36,32,47,45)(28,42,41,52,43,39,51,49);(5,8,10,9)(6,12,13,17)(7,15,16,18)(11,19,14,20)(24,36)(28,43)(32,35)(34,47)(39,42)(41,51)(45,48)(49,52);(5,9,10,8)(6,12,13,17)(7,15,16,18)(11,20,14,19)(21,24,37,47,26,36,25,34)(22,28,44,51,30,43,29,41)(23,32,46,48,33,35,31,45)(27,39,50,52,40,42,38,49)Polycyclic presentation:
2^9|a^2=fi,b^2=f,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,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, 6719119);