Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993947 |
| 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×#449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1741 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)(7,10,13,17)(8,15,16,11)(14,20,19,18)(21,22,28,34)(23,35,39,50)(24,30,25,31)(26,33,27,32)(29,43,46,38)(36,51,40,45)(37,47,52,44)(41,49,42,48);(7,13)(8,16)(10,17)(11,15)(21,23,27,38,28,39,26,43)(22,29,33,35,34,46,32,50)(24,40,41,37,25,36,42,52)(30,47,48,45,31,44,49,51);(2,4)(5,7)(6,10)(8,14)(9,13)(11,18)(12,17)(15,20)(16,19)(21,24)(22,30)(23,36)(25,28)(26,42)(27,41)(29,44)(31,34)(32,49)(33,48)(35,51)(37,43)(38,52)(39,40)(45,50)(46,47);(7,13)(10,17)(14,19)(18,20)(21,25,28,24)(22,31,34,30)(23,37,39,52)(26,42,27,41)(29,45,46,51)(32,49,33,48)(35,47,50,44)(36,38,40,43);(5,8,9,16)(6,11,12,15)(7,14,13,19)(10,18,17,20)(21,26,28,27)(22,32,34,33)(23,38,39,43)(24,41,25,42)(29,35,46,50)(30,48,31,49)(36,37,40,52)(44,45,47,51)Polycyclic presentation:
2^9|a^2=h,b^2=f,d^2=i,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=eh,d^c=dh,e^c=ei,f^d=fi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7074213);