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 | C22×D4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1646 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)(21,22,25,29)(23,31,34,27)(24,36,38,28)(26,30,33,40)(32,43,45,39)(35,41,48,44)(37,49,47,42)(46,50,52,51);(5,7)(6,11)(8,16)(9,18)(10,17)(12,19)(13,20)(14,15)(23,32)(24,37)(26,33)(27,39)(28,42)(30,40)(31,43)(34,45)(35,48)(36,49)(38,47)(41,44);(1,2,3,4)(5,8,10,9)(6,12,14,13)(7,16,17,18)(11,19,15,20)(21,23,33,45,25,34,26,32)(22,27,40,43,29,31,30,39)(24,35,47,52,38,48,37,46)(28,41,49,51,36,44,42,50);(5,8,10,9)(6,13,14,12)(7,16,17,18)(11,20,15,19)(23,34)(27,31)(32,45)(35,48)(39,43)(41,44)(46,52)(50,51);(5,9,10,8)(6,13,14,12)(7,16,17,18)(11,19,15,20)(21,24,26,37,25,38,33,47)(22,28,30,42,29,36,40,49)(23,35,32,46,34,48,45,52)(27,41,39,50,31,44,43,51)Polycyclic presentation:
2^9|a^2=fi,c^2=fgh,d^2=f,e^2=fh,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, 6724548);