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 | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1670 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)(8,16)(11,17)(15,20)(18,19)(21,22,28,44,27,33,26,32)(23,34,37,29,39,48,38,49)(24,40,41,50,36,30,42,47)(25,31,45,52,43,46,35,51);(5,6,9,12)(7,13,14,10)(8,17,16,11)(15,18,20,19)(21,23,27,39)(22,29,33,49)(24,35,36,45)(25,41,43,42)(26,37,28,38)(30,46,40,31)(32,48,44,34)(47,52,50,51);(2,4)(5,7)(6,10)(8,15)(9,14)(11,18)(12,13)(16,20)(17,19)(21,24)(22,30)(23,35)(25,37)(26,42)(27,36)(28,41)(29,46)(31,49)(32,50)(33,40)(34,52)(38,43)(39,45)(44,47)(48,51);(7,14)(10,13)(15,20)(18,19)(21,25,28,45,27,43,26,35)(22,31,44,52,33,46,32,51)(23,36,37,42,39,24,38,41)(29,47,48,40,49,50,34,30);(5,8,9,16)(6,11,12,17)(7,15,14,20)(10,18,13,19)(21,26,27,28)(22,32,33,44)(23,37,39,38)(24,41,36,42)(25,35,43,45)(29,48,49,34)(30,47,40,50)(31,51,46,52)Polycyclic presentation:
2^9|a^2=fg,b^2=hi,d^2=fi,e^2=hi,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, 7090835);