Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667556 |
| 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 | #42908 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1845 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)(7,15)(8,12)(9,14)(10,13)(11,17)(16,20)(18,19);(5,7)(6,11)(8,16)(9,18)(10,17)(12,19)(13,15)(14,20)(21,22)(23,32)(24,28)(25,29)(26,31)(27,34)(30,40)(33,48)(35,47)(36,50)(37,44)(38,46)(39,42)(41,49)(43,51)(45,52);(2,4)(5,8,10,9)(6,12,13,14)(7,16,17,18)(11,19,15,20)(21,23,26,35)(22,27,31,42)(24,36,38,33)(25,39,40,34)(28,43,46,41)(29,32,30,47)(37,49,52,51)(44,50,45,48);(6,13)(11,15)(12,14)(19,20)(21,24,26,38)(22,28,31,46)(23,33,35,36)(25,37,40,52)(27,41,42,43)(29,44,30,45)(32,48,47,50)(34,49,39,51);(5,9,10,8)(6,14,13,12)(7,16,17,18)(11,19,15,20)(22,29)(25,40)(27,32)(28,44)(30,31)(34,39)(37,52)(41,48)(42,47)(43,50)(45,46)(49,51)Polycyclic presentation:
2^9|c^2=fi,d^2=i,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6581236);