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 | |
1625 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)(5,6,10,8)(7,11,13,14)(9,16,17,12)(15,20,19,18)(21,22)(23,29)(24,30)(25,28)(26,27)(31,34)(32,36)(33,35)(37,38,41,44)(39,45,47,42)(40,48,49,43)(46,50,52,51);(7,13)(11,14)(15,19)(18,20)(22,27)(23,30)(25,33)(32,34)(37,38,41,44)(39,42,47,45)(40,48,49,43)(46,51,52,50);(5,7,10,13)(6,11,8,14)(9,15,17,19)(12,18,16,20)(21,23)(22,24)(25,31)(26,30)(27,29)(28,32)(33,36)(34,35)(37,38)(39,45)(40,43)(41,44)(42,47)(46,51)(48,49)(50,52);(1,2)(3,4)(5,8,10,6)(7,14,13,11)(9,16,17,12)(15,20,19,18)(21,24)(22,23)(25,34)(26,29)(27,30)(28,36)(31,35)(32,33)(37,39)(38,42)(40,46)(41,47)(43,50)(44,45)(48,51)(49,52);(5,9)(6,12)(7,15)(8,16)(10,17)(11,18)(13,19)(14,20)(21,25,26,33)(22,28,27,35)(23,31,30,36)(24,32,29,34)(37,40,41,49)(38,43,44,48)(39,46,47,52)(42,50,45,51)Polycyclic presentation:
2^9|a^2=fh,b^2=fg,c^2=gh,d^2=gh,e^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, 5527206);