Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667772 |
| 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 | |
1719 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)(5,6,10,13)(7,14,16,11)(8,17,9,12)(15,19,18,20)(21,22,27,31)(23,28,36,45)(24,29,40,48)(25,30,41,26)(32,43,49,51)(33,35,44,34)(37,46,50,52)(38,39,47,42);(5,7,10,16)(6,11,13,14)(8,15,9,18)(12,19,17,20)(21,23,27,36)(22,28,31,45)(24,37,40,50)(25,33,41,44)(26,34,30,35)(29,46,48,52)(32,42,49,39)(38,51,47,43);(2,4)(21,24,27,40)(22,29,31,48)(23,32,36,49)(25,38,41,47)(26,42,30,39)(28,43,45,51)(33,46,44,52)(34,37,35,50);(5,8,10,9)(6,12,13,17)(7,15,16,18)(11,19,14,20)(21,25)(22,30)(23,33)(24,38)(26,31)(27,41)(28,35)(29,39)(32,46)(34,45)(36,44)(37,51)(40,47)(42,48)(43,50)(49,52);(5,9,10,8)(6,12,13,17)(7,15,16,18)(11,20,14,19)(23,34)(25,41)(26,30)(28,33)(32,37)(35,36)(38,47)(39,42)(43,46)(44,45)(49,50)(51,52)Polycyclic presentation:
2^9|a^2=fi,b^2=fi,c^2=i,d^2=f,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6584716);