Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806623 |
| 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 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1686 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)(21,22,26,30)(23,27,33,39)(24,28,36,42)(25,29,32,38)(31,37,43,47)(34,40,46,50)(35,41,45,49)(44,48,51,52);(5,7,10,16)(6,11,13,14)(8,15,9,18)(12,19,17,20)(23,31)(24,35)(25,32)(27,37)(28,41)(29,38)(33,43)(34,46)(36,45)(39,47)(40,50)(42,49);(2,4)(21,23,32,43,26,33,25,31)(22,27,38,47,30,39,29,37)(24,34,45,51,36,46,35,44)(28,40,49,52,42,50,41,48);(5,8,10,9)(6,12,13,17)(7,15,16,18)(11,19,14,20)(23,33)(27,39)(31,43)(34,46)(37,47)(40,50)(44,51)(48,52);(5,9,10,8)(6,12,13,17)(7,15,16,18)(11,20,14,19)(21,24,32,45,26,36,25,35)(22,28,38,49,30,42,29,41)(23,34,43,51,33,46,31,44)(27,40,47,52,39,50,37,48)Polycyclic presentation:
2^9|a^2=fi,b^2=f,c^2=hi,d^2=f,e^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6718920);