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 | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1688 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)(5,6,9,12)(7,13,15,10)(8,16,17,11)(14,18,20,19)(21,22,25,29)(23,31,33,27)(24,35,37,28)(26,30,38,42)(32,43,45,39)(34,40,47,44)(36,48,49,41)(46,50,52,51);(5,7)(6,10)(8,14)(9,15)(11,18)(12,13)(16,19)(17,20)(23,32)(24,36)(26,38)(27,39)(28,41)(30,42)(31,43)(33,45)(34,47)(35,48)(37,49)(40,44);(2,4)(21,23,26,32,25,33,38,45)(22,27,30,39,29,31,42,43)(24,34,36,46,37,47,49,52)(28,40,41,50,35,44,48,51);(5,8,9,17)(6,11,12,16)(7,14,15,20)(10,18,13,19)(23,33)(27,31)(32,45)(34,47)(39,43)(40,44)(46,52)(50,51);(7,15)(10,13)(14,20)(18,19)(21,24,26,36,25,37,38,49)(22,28,30,41,29,35,42,48)(23,34,32,46,33,47,45,52)(27,40,39,50,31,44,43,51)Polycyclic presentation:
2^9|a^2=fi,c^2=h,d^2=f,e^2=h,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, 6715789);