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 | C22×D4 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1628 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)(8,17)(9,12)(10,13)(11,16)(15,19)(18,20)(21,22,27,33)(23,34,37,29)(24,30,35,44)(25,39,28,31)(26,32,42,49)(36,50,51,45)(38,46,41,48)(40,52,43,47);(5,7)(6,11)(8,15)(9,18)(10,16)(12,19)(13,14)(17,20)(21,23,27,37)(22,29,33,34)(24,25,35,28)(26,38,42,41)(30,31,44,39)(32,48,49,46)(36,43,51,40)(45,47,50,52);(1,2,3,4)(21,24,28,23,27,35,25,37)(22,30,31,34,33,44,39,29)(26,41,43,51,42,38,40,36)(32,48,47,45,49,46,52,50);(5,8,10,9)(6,12,13,17)(7,15,16,18)(11,19,14,20)(21,25,27,28)(22,31,33,39)(23,35,37,24)(26,40,42,43)(29,44,34,30)(32,47,49,52)(36,41,51,38)(45,48,50,46);(5,9,10,8)(6,12,13,17)(7,15,16,18)(11,20,14,19)(21,26,28,43,27,42,25,40)(22,32,31,47,33,49,39,52)(23,36,35,41,37,51,24,38)(29,45,30,46,34,50,44,48)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=gh,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, 6723781);