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 | #28552 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1723 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)(5,6,9,12)(7,10,14,18)(8,11,16,20)(13,17,15,19)(21,22,25,31,26,30,36,40)(23,28,34,42,33,44,39,27)(24,29,37,46,38,45,47,51)(32,43,35,49,48,52,50,41);(5,7)(6,10)(8,15)(9,14)(11,19)(12,18)(13,16)(17,20)(21,23,26,33)(22,27,30,42)(24,35,38,50)(25,39,36,34)(28,31,44,40)(29,43,45,52)(32,47,48,37)(41,51,49,46);(2,4)(5,8,9,16)(6,11,12,20)(7,13,14,15)(10,17,18,19)(21,23)(22,28)(24,32)(25,34)(26,33)(27,40)(29,43)(30,44)(31,42)(35,37)(36,39)(38,48)(41,51)(45,52)(46,49)(47,50);(8,16)(11,20)(13,15)(17,19)(21,24,36,47,26,38,25,37)(22,29,40,51,30,45,31,46)(23,32,39,50,33,48,34,35)(27,41,44,52,42,49,28,43);(6,12)(7,14)(11,20)(13,15)(22,30)(23,33)(29,45)(31,40)(32,48)(34,39)(35,50)(46,51)Polycyclic presentation:
2^9|a^2=fhi,b^2=i,c^2=hi,d^2=fi,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dhi,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7011590);