Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500434 |
| 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 | #35210 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1768 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)(7,10,13,16)(8,11,15,18)(14,17,19,20)(21,22,27,31)(23,28,33,43)(24,35,39,51)(25,30,26,41)(29,37,47,38)(32,48,36,50)(34,44,40,45)(42,52,46,49);(2,4)(5,7)(6,10)(8,14)(9,13)(11,17)(12,16)(15,19)(18,20)(21,23)(22,28)(24,36)(25,40)(26,34)(27,33)(29,46)(30,45)(31,43)(32,39)(35,50)(37,49)(38,52)(41,44)(42,47)(48,51);(7,13)(8,15)(10,16)(11,18)(21,24,26,38,27,39,25,37)(22,29,30,35,31,47,41,51)(23,32,34,49,33,36,40,52)(28,42,45,48,43,46,44,50);(5,8,9,15)(6,11,12,18)(7,14,13,19)(10,17,16,20)(23,33)(28,43)(32,36)(34,40)(42,46)(44,45)(48,50)(49,52);(7,13)(10,16)(14,19)(17,20)(21,25,27,26)(22,30,31,41)(23,34,33,40)(24,37,39,38)(28,44,43,45)(29,35,47,51)(32,49,36,52)(42,50,46,48)Polycyclic presentation:
2^9|a^2=h,c^2=g,d^2=hi,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dhi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6433803);