Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682380 |
| 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×#466 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1746 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(5,6)(7,13)(8,12)(9,11)(10,15)(14,20)(16,18)(17,19)(21,22)(23,31)(24,35)(25,29)(26,39)(27,33)(28,37)(30,38)(32,45)(34,41)(36,50)(40,52)(42,51)(43,47)(44,48)(46,49);(1,2)(3,4)(5,7,9,15)(6,10,11,13)(8,16,17,14)(12,20,19,18)(21,23)(22,27)(24,32)(25,33)(26,34)(28,40)(29,31)(30,41)(35,49)(36,46)(37,47)(38,48)(39,44)(42,43)(45,50)(51,52);(2,4)(21,24,26,37,25,36,38,51)(22,28,30,35,29,42,39,50)(23,32,34,47,33,46,48,52)(27,40,41,49,31,43,44,45);(6,11)(10,13)(12,19)(18,20)(22,29)(23,33)(28,42)(30,39)(32,46)(34,48)(35,50)(47,52);(2,4)(5,8,9,17)(6,12,11,19)(7,14,15,16)(10,18,13,20)(23,33)(27,31)(32,46)(34,48)(40,43)(41,44)(45,49)(47,52)Polycyclic presentation:
2^9|b^2=fi,c^2=g,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6593032);