Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #6132643 |
| 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 | |
1338 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,4,6)(3,5,7,8)(10,14)(11,17)(16,21)(19,23)(25,26,30,27)(28,31,35,33)(29,34,37,32)(36,39,40,38)(41,42)(43,49)(44,52)(45,46)(47,50)(48,53)(51,55)(54,56);(1,3,4,7)(2,5,6,8)(10,14)(11,17)(12,20)(22,24)(26,27)(28,35)(32,34)(36,40)(42,46)(43,50)(44,53)(55,56);(1,3)(2,5)(4,7)(6,8)(9,10)(11,18)(12,16)(13,14)(15,17)(19,24)(20,21)(22,23)(25,27,30,26)(28,31,35,33)(29,32,37,34)(36,39,40,38)(41,43)(42,47)(44,54)(45,50)(46,49)(48,56)(51,53)(52,55);(9,11)(10,15)(12,19)(13,17)(14,18)(16,22)(20,23)(21,24)(25,28)(26,31)(27,33)(29,36)(30,35)(32,38)(34,39)(37,40)(41,42)(43,47)(44,48)(45,46)(49,50)(51,55)(52,53)(54,56);(1,2)(3,5)(4,6)(7,8)(9,12,13,20)(10,16,14,21)(11,19,17,23)(15,22,18,24)(25,29,30,37)(26,32,27,34)(28,36,35,40)(31,38,33,39)(41,44)(42,48)(43,51)(45,53)(46,52)(47,55)(49,56)(50,54)Polycyclic presentation:
2^9|a^2=gh,b^2=fhi,c^2=fgi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dfi,e^b=eg,d^c=dfg,e^c=ef62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6176786);