Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470374 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3212 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2049 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)(7,10)(8,15)(9,11)(12,16)(13,17)(14,20)(18,19)(21,22)(23,32)(24,29)(25,30)(26,41)(27,31)(28,34)(33,48)(35,43)(36,50)(37,46)(38,47)(39,52)(40,44)(42,49)(45,51);(5,7)(6,10)(8,14)(9,13)(11,17)(12,18)(15,20)(16,19)(21,23,26,34,27,35,25,40)(22,28,30,32,31,44,41,43)(24,36,39,51,38,33,37,49)(29,45,46,50,47,42,52,48);(1,2,3,4)(6,11)(7,13)(12,15)(14,19)(21,24,27,38)(22,29,31,47)(23,33,35,36)(25,37,26,39)(28,42,44,45)(30,46,41,52)(32,48,43,50)(34,49,40,51);(7,13)(10,17)(14,19)(18,20)(21,25,27,26)(22,30,31,41)(23,34,35,40)(24,37,38,39)(28,43,44,32)(29,46,47,52)(33,49,36,51)(42,50,45,48);(5,8,9,16)(6,12,11,15)(7,14,13,19)(10,18,17,20)(22,31)(24,38)(28,44)(30,41)(32,43)(33,36)(37,39)(49,51)Polycyclic presentation:
2^9|b^2=f,c^2=gh,d^2=i,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6405951);