Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806671 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1620 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)(25,38)(30,45)(34,48)(37,40)(39,51)(43,49)(46,52)(47,50);(5,7,10,16)(6,11,13,14)(8,15,9,18)(12,19,17,20)(21,22)(23,32)(24,29)(25,39)(26,41)(27,31)(28,33)(30,40)(34,49)(35,42)(36,44)(37,45)(38,51)(43,48)(46,47)(50,52);(2,4)(21,23,26,33,27,35,24,36)(22,28,29,32,31,44,41,42)(25,34,40,47,38,48,37,50)(30,43,39,46,45,49,51,52);(5,8,10,9)(6,12,13,17)(7,15,16,18)(11,19,14,20)(21,24,27,26)(22,29,31,41)(23,33,35,36)(25,37,38,40)(28,42,44,32)(30,39,45,51)(34,47,48,50)(43,52,49,46);(5,9,10,8)(6,12,13,17)(7,15,16,18)(11,20,14,19)(21,25,24,37,27,38,26,40)(22,30,41,51,31,45,29,39)(23,34,36,50,35,48,33,47)(28,43,42,52,44,49,32,46)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=h,d^2=fi,e^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6718951);