Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533195 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3565 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2757 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)(21,22,26,31,27,32,25,30)(23,33,35,42,36,46,40,28)(24,29,39,45,37,43,38,44)(34,47,49,51,48,52,50,41);(5,6,9,12)(7,13,15,10)(8,11,16,18)(14,19,20,17)(21,23)(22,28)(24,34)(25,35)(26,40)(27,36)(29,41)(30,33)(31,46)(32,42)(37,48)(38,49)(39,50)(43,51)(44,47)(45,52);(2,4)(5,7,9,15)(6,10,12,13)(8,14,16,20)(11,17,18,19)(21,24)(22,29)(23,34)(25,38)(26,39)(27,37)(28,41)(30,44)(31,45)(32,43)(33,47)(35,49)(36,48)(40,50)(42,51)(46,52);(5,8)(6,11)(7,14)(9,16)(10,17)(12,18)(13,19)(15,20)(24,37)(29,43)(34,48)(38,39)(41,51)(44,45)(47,52)(49,50);(8,16)(11,18)(14,20)(17,19)(21,25,27,26)(22,30,32,31)(23,35,36,40)(24,38,37,39)(28,33,42,46)(29,44,43,45)(34,49,48,50)(41,47,51,52)Polycyclic presentation:
2^9|a^2=f,b^2=h,c^2=h,e^2=i,f^2=i,b^a=bf,c^a=cg,c^b=ch,f^b=fi,d^c=di,e^d=eh77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6479856);