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 | |
1669 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)(21,22,27,32)(23,28,35,43)(24,29,34,42)(25,30,36,44)(26,31,40,48)(33,41,50,52)(37,45,39,47)(38,46,49,51);(5,7)(6,11)(8,16)(9,18)(10,17)(12,19)(13,20)(14,15)(21,23,27,35)(22,28,32,43)(24,36,34,25)(26,39,40,37)(29,44,42,30)(31,47,48,45)(33,49,50,38)(41,51,52,46);(1,2,3,4)(5,8,10,9)(6,12,14,13)(7,16,17,18)(11,19,15,20)(21,24,25,23,27,34,36,35)(22,29,30,28,32,42,44,43)(26,37,38,33,40,39,49,50)(31,45,46,41,48,47,51,52);(5,8,10,9)(6,13,14,12)(7,16,17,18)(11,20,15,19)(21,25,27,36)(22,30,32,44)(23,24,35,34)(26,38,40,49)(28,29,43,42)(31,46,48,51)(33,37,50,39)(41,45,52,47);(5,9,10,8)(6,13,14,12)(7,16,17,18)(11,19,15,20)(21,26,36,49,27,40,25,38)(22,31,44,51,32,48,30,46)(23,33,24,37,35,50,34,39)(28,41,29,45,43,52,42,47)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=fgh,d^2=fi,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, 6724535);