Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5956507 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
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 | |
1346 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(3,7)(6,8)(9,10,14,11)(12,17,19,15)(13,18,21,16)(20,22,24,23)(25,26)(27,28)(29,37)(30,31)(32,38)(33,36)(34,40)(35,39)(42,46)(44,52)(47,49)(50,56);(2,5)(3,7)(12,19)(15,17)(20,24)(22,23)(26,31)(27,33)(29,38)(35,40)(41,42,45,46)(43,49,51,47)(44,53,52,48)(50,54,56,55);(1,2)(3,6)(4,5)(7,8)(9,11,14,10)(12,17,19,15)(13,18,21,16)(20,23,24,22)(25,27,30,33)(26,28,31,36)(29,34,38,39)(32,35,37,40)(41,42)(43,47)(44,48)(45,46)(49,51)(50,54)(52,53)(55,56);(1,3)(2,6)(4,7)(5,8)(9,12,14,19)(10,15,11,17)(13,20,21,24)(16,22,18,23)(25,28)(26,27)(29,35)(30,36)(31,33)(32,34)(37,39)(38,40)(41,43,45,51)(42,47,46,49)(44,50,52,56)(48,54,53,55);(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,29)(26,32)(27,34)(28,35)(30,38)(31,37)(33,39)(36,40)(41,44,45,52)(42,48,46,53)(43,50,51,56)(47,54,49,55)Polycyclic presentation:
2^9|a^2=fhi,b^2=g,c^2=ghi,d^2=fghi,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dgi,e^b=ef,d^c=dfg62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6029420);