Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667580 |
| 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×#553 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1734 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)(7,13)(8,12)(9,11)(10,15)(14,20)(16,18)(17,19)(21,22,26,27)(23,31,34,28)(24,29,37,41)(25,38,40,30)(32,35,48,42)(33,43,50,46)(36,45,39,44)(47,52,49,51);(5,7,9,15)(6,10,11,13)(8,16,17,14)(12,20,19,18)(21,23,26,34)(22,28,27,31)(24,35,37,42)(25,33,40,50)(29,32,41,48)(30,43,38,46)(36,51,39,52)(44,47,45,49);(2,4)(21,24,26,37)(22,29,27,41)(23,32,34,48)(25,39,40,36)(28,42,31,35)(30,45,38,44)(33,49,50,47)(43,51,46,52);(6,11)(10,13)(12,19)(18,20)(21,25,26,40)(22,30,27,38)(23,33,34,50)(24,36,37,39)(28,43,31,46)(29,44,41,45)(32,47,48,49)(35,51,42,52);(5,8,9,17)(6,12,11,19)(7,14,15,16)(10,18,13,20)(22,27)(23,28)(24,37)(30,38)(31,34)(32,35)(33,43)(36,39)(42,48)(46,50)(47,51)(49,52)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=i,d^2=i,e^2=fi,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=ei,h^c=hi,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6580673);