Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806839 |
| 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 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1644 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)(5,6,10,14)(7,15,17,11)(8,12,9,13)(16,20,18,19)(21,22)(23,28)(24,36)(25,30)(26,31)(27,32)(29,35)(33,42)(34,43)(37,46)(38,50)(39,44)(40,45)(41,47)(48,51)(49,52);(5,7)(6,11)(8,16)(9,18)(10,17)(12,19)(13,20)(14,15)(21,23,26,34)(22,28,31,43)(24,37,35,27)(25,39,40,38)(29,32,36,46)(30,44,45,50)(33,48,49,41)(42,51,52,47);(1,2,3,4)(5,8,10,9)(6,12,14,13)(7,16,17,18)(11,19,15,20)(21,24,27,23,26,35,37,34)(22,29,32,43,31,36,46,28)(25,38,41,33,40,39,48,49)(30,44,47,52,45,50,51,42);(5,8,10,9)(6,13,14,12)(7,16,17,18)(11,20,15,19)(25,40)(30,45)(33,49)(38,39)(41,48)(42,52)(44,50)(47,51);(5,9,10,8)(6,13,14,12)(7,16,17,18)(11,19,15,20)(21,25,37,48,26,40,27,41)(22,30,46,51,31,45,32,47)(23,33,24,38,34,49,35,39)(28,42,36,50,43,52,29,44)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=fghi,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,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6724700);