Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1636 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)(6,10)(7,13)(12,17)(15,19)(21,22,25,26)(23,29,31,27)(24,32,33,28)(30,34,36,35)(37,38)(39,43)(40,42)(41,48)(44,47)(45,50)(46,52)(49,51);(8,16)(12,17)(15,19)(18,20)(22,26)(27,29)(28,32)(34,35)(37,39,42,47)(38,43,40,44)(41,49,50,46)(45,52,48,51);(5,6,9,10)(7,14,13,11)(8,12,16,17)(15,20,19,18)(21,23)(22,27)(24,30)(25,31)(26,29)(28,34)(32,35)(33,36)(37,39)(38,44)(40,43)(41,46)(42,47)(45,52)(48,51)(49,50);(1,2)(3,4)(5,7,9,13)(6,11,10,14)(8,15,16,19)(12,18,17,20)(21,24,25,33)(22,28,26,32)(23,30,31,36)(27,34,29,35)(37,40,42,38)(39,44,47,43)(41,48,50,45)(46,52,49,51);(5,8)(6,12)(7,15)(9,16)(10,17)(11,18)(13,19)(14,20)(21,22)(23,27)(24,32)(25,26)(28,33)(29,31)(30,35)(34,36)(37,41)(38,45)(39,46)(40,48)(42,50)(43,51)(44,52)(47,49)Polycyclic presentation:
2^9|a^2=f,b^2=h,c^2=fgh,d^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5519464);