Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667772 |
| 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 | #42674 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1737 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)(7,13)(8,12)(9,11)(10,15)(14,20)(16,18)(17,19)(21,22,25,29)(23,31,35,27)(24,36,38,28)(26,30,40,46)(32,43,34,41)(33,42,37,44)(39,49,50,45)(47,51,48,52);(5,7,9,15)(6,10,11,13)(8,16,17,14)(12,20,19,18)(21,23,25,35)(22,27,29,31)(24,37,38,33)(26,32,40,34)(28,44,36,42)(30,41,46,43)(39,48,50,47)(45,52,49,51);(1,2,3,4)(23,32)(24,38)(26,40)(27,41)(28,36)(30,46)(31,43)(33,48)(34,35)(37,47)(42,52)(44,51);(6,11)(10,13)(12,19)(18,20)(21,24,25,38)(22,28,29,36)(23,33,35,37)(26,39,40,50)(27,42,31,44)(30,45,46,49)(32,47,34,48)(41,51,43,52);(5,8,9,17)(6,12,11,19)(7,14,15,16)(10,18,13,20)(23,34)(26,40)(27,43)(30,46)(31,41)(32,35)(33,48)(37,47)(39,50)(42,52)(44,51)(45,49)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=g,d^2=i,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,d^b=di,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6593443);