Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996375 |
| 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 | #45136 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1792 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)(5,6,9,12)(7,10,14,18)(8,11,16,20)(13,17,15,19)(21,22,28,40,27,31,26,32)(23,33,25,29,36,46,35,43)(24,30,41,50,38,45,39,44)(34,47,37,42,48,52,49,51);(5,7)(6,10)(8,15)(9,14)(11,19)(12,18)(13,16)(17,20)(21,23,27,36)(22,29,31,43)(24,34,38,48)(25,28,35,26)(30,42,45,51)(32,46,40,33)(37,41,49,39)(44,52,50,47);(1,2,3,4)(5,8)(6,11)(7,13)(9,16)(10,17)(12,20)(14,15)(18,19)(21,24,27,38)(22,30,31,45)(23,34,36,48)(25,37,35,49)(26,39,28,41)(29,42,43,51)(32,44,40,50)(33,47,46,52);(8,16)(11,20)(13,15)(17,19)(21,25,26,23,27,35,28,36)(22,29,32,33,31,43,40,46)(24,37,39,34,38,49,41,48)(30,42,44,47,45,51,50,52);(6,12)(7,14)(11,20)(13,15)(22,31)(24,38)(29,43)(32,40)(33,46)(34,48)(37,49)(39,41)Polycyclic presentation:
2^9|a^2=fhi,b^2=i,c^2=gi,d^2=f,f^2=i,b^a=bf,c^a=cg,e^a=ehi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6949523);