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 | #2540 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1746 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(2,4)(5,6)(7,14)(8,12)(9,18)(10,13)(11,17)(15,20)(16,19)(21,22,27,33)(23,34,37,29)(24,38,35,30)(25,40,28,31)(26,42,44,32)(36,46,51,50)(39,47,43,49)(41,48,45,52);(5,7)(6,11)(8,15)(9,16)(10,17)(12,19)(13,14)(18,20)(21,23,27,37)(22,29,33,34)(24,25,35,28)(26,39,44,43)(30,40,38,31)(32,47,42,49)(36,45,51,41)(46,48,50,52);(1,2,3,4)(5,8,10,9)(6,12,13,18)(7,15,17,16)(11,19,14,20)(21,24,28,23,27,35,25,37)(22,30,31,29,33,38,40,34)(26,43,45,51,44,39,41,36)(32,49,48,50,42,47,52,46);(5,9,10,8)(6,12,13,18)(7,16,17,15)(11,19,14,20)(21,25,27,28)(22,31,33,40)(23,35,37,24)(26,41,44,45)(29,30,34,38)(32,48,42,52)(36,43,51,39)(46,47,50,49);(7,17)(11,14)(15,16)(19,20)(21,26,25,41,27,44,28,45)(22,32,40,52,33,42,31,48)(23,36,24,39,37,51,35,43)(29,46,30,47,34,50,38,49)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fgh,d^2=f,e^2=hi,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, 6722752);