Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| 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 | #392 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1694 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)(21,22,26,31,27,30,41,47)(23,32,35,43,34,48,25,28)(24,36,40,52,39,29,44,46)(33,50,51,42,37,49,38,45);(5,6,9,12)(7,13,14,10)(8,17,16,11)(15,18,20,19)(21,23,27,34)(22,28,30,43)(24,37,39,33)(25,41,35,26)(29,45,36,42)(31,48,47,32)(38,40,51,44)(46,49,52,50);(2,4)(5,7)(6,10)(8,15)(9,14)(11,18)(12,13)(16,20)(17,19)(21,24,27,39)(22,29,30,36)(23,33,34,37)(25,38,35,51)(26,40,41,44)(28,42,43,45)(31,46,47,52)(32,49,48,50);(7,14)(10,13)(15,20)(18,19)(21,25,26,23,27,35,41,34)(22,28,31,32,30,43,47,48)(24,38,40,33,39,51,44,37)(29,42,46,49,36,45,52,50);(5,8,9,16)(6,11,12,17)(7,15,14,20)(10,18,13,19)(22,30)(23,34)(24,39)(25,35)(31,47)(40,44)(42,45)(49,50)Polycyclic presentation:
2^9|a^2=f,b^2=h,c^2=i,d^2=f,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=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7088470);