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)(7,13)(8,15)(11,16)(12,17)(21,22,25,28)(23,29,31,26)(24,27,32,34)(30,35,36,33)(38,42)(39,45)(44,47)(46,48);(6,10)(11,16)(12,17)(18,20)(24,32)(27,34)(30,36)(33,35)(37,38,41,42)(39,43,45,50)(40,47,49,44)(46,52,48,51);(5,6)(7,11)(8,12)(9,10)(13,16)(14,18)(15,17)(19,20)(21,23,25,31)(22,26,28,29)(24,30,32,36)(27,33,34,35)(37,38)(39,43)(40,44)(41,42)(45,50)(46,51)(47,49)(48,52);(1,2)(3,4)(5,7,9,13)(6,11,10,16)(8,14,15,19)(12,18,17,20)(21,22)(23,29)(24,27)(25,28)(26,31)(30,35)(32,34)(33,36)(37,39,41,45)(38,43,42,50)(40,48,49,46)(44,52,47,51);(5,8)(6,12)(7,14)(9,15)(10,17)(11,18)(13,19)(16,20)(21,24)(22,27)(23,30)(25,32)(26,33)(28,34)(29,35)(31,36)(37,40,41,49)(38,44,42,47)(39,46,45,48)(43,51,50,52)Polycyclic presentation:
2^9|a^2=fg,b^2=f,c^2=fg,d^2=h,e^2=f,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, 5521306);