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 | C22×D8 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1683 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(2,4)(5,6,10,14)(7,15,17,11)(8,12,9,13)(16,20,18,19)(21,22,26,30)(23,27,32,38)(24,28,36,42)(25,29,35,41)(31,37,43,47)(33,39,45,49)(34,40,46,50)(44,48,51,52);(5,7)(6,11)(8,16)(9,18)(10,17)(12,19)(13,20)(14,15)(23,31)(24,34)(25,35)(27,37)(28,40)(29,41)(32,43)(33,45)(36,46)(38,47)(39,49)(42,50);(1,2,3,4)(5,8,10,9)(6,12,14,13)(7,16,17,18)(11,19,15,20)(21,23,25,31,26,32,35,43)(22,27,29,37,30,38,41,47)(24,33,34,44,36,45,46,51)(28,39,40,48,42,49,50,52);(5,8,10,9)(6,13,14,12)(7,16,17,18)(11,20,15,19)(23,32)(27,38)(31,43)(33,45)(37,47)(39,49)(44,51)(48,52);(5,9,10,8)(6,13,14,12)(7,16,17,18)(11,19,15,20)(21,24,35,46,26,36,25,34)(22,28,41,50,30,42,29,40)(23,33,43,51,32,45,31,44)(27,39,47,52,38,49,37,48)Polycyclic presentation:
2^9|a^2=fi,c^2=fgh,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, 6724529);