Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682380 |
| 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×#466 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1737 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(5,6)(7,13)(8,12)(9,11)(10,15)(14,20)(16,18)(17,19)(21,22,27,31)(23,28,34,44)(24,35,38,49)(25,30,26,40)(29,36,45,37)(32,46,48,51)(33,42,39,43)(41,50,52,47);(1,2)(3,4)(5,7,9,15)(6,10,11,13)(8,16,17,14)(12,20,19,18)(21,23)(22,28)(24,32)(25,39)(26,33)(27,34)(29,41)(30,43)(31,44)(35,46)(36,50)(37,47)(38,48)(40,42)(45,52)(49,51);(2,4)(21,24,25,36,27,38,26,37)(22,29,40,49,31,45,30,35)(23,32,39,50,34,48,33,47)(28,41,42,51,44,52,43,46);(6,11)(10,13)(12,19)(18,20)(21,25,27,26)(22,30,31,40)(23,33,34,39)(24,36,38,37)(28,42,44,43)(29,35,45,49)(32,47,48,50)(41,51,52,46);(2,4)(5,8,9,17)(6,12,11,19)(7,14,15,16)(10,18,13,20)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=gi,d^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6593029);