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 | |
1647 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)(23,34)(24,30)(25,42)(26,32)(27,33)(28,31)(29,38)(35,51)(36,46)(37,49)(39,52)(40,47)(41,50)(43,45)(44,48);(5,7,10,16)(6,11,13,14)(8,15,9,18)(12,19,17,20)(21,23)(22,29)(24,39)(25,36)(26,41)(27,38)(28,44)(30,32)(31,46)(33,34)(35,40)(37,43)(42,48)(45,47)(49,51)(50,52);(2,4)(21,24,28,40,27,41,25,37)(22,30,31,47,33,50,42,49)(23,35,36,39,38,43,44,26)(29,45,48,32,34,51,46,52);(5,8,10,9)(6,12,13,17)(7,15,16,18)(11,19,14,20)(21,25,27,28)(22,31,33,42)(23,36,38,44)(24,40,41,37)(26,43,39,35)(29,46,34,48)(30,49,50,47)(32,51,52,45);(5,9,10,8)(6,12,13,17)(7,15,16,18)(11,20,14,19)(21,26,28,35,27,39,25,43)(22,32,31,51,33,52,42,45)(23,37,36,24,38,40,44,41)(29,47,48,50,34,49,46,30)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=di,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6718976);