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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1240 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 56 points:
(2,5)(3,7)(10,14)(11,17)(16,21)(19,23)(25,26,30,27)(28,33,35,31)(29,32,37,34)(36,39,40,38)(41,42)(43,48)(44,50)(45,47)(46,53)(49,54)(51,56)(52,55);(2,5)(6,8)(10,14)(15,18)(16,21)(22,24)(26,27)(28,35)(29,37)(38,39)(41,43,47,44)(42,48,45,50)(46,51,54,52)(49,55,53,56);(9,10)(11,18)(12,16)(13,14)(15,17)(19,24)(20,21)(22,23)(25,27,30,26)(28,31,35,33)(29,34,37,32)(36,38,40,39)(41,44,47,43)(42,48,45,50)(46,52,54,51)(49,55,53,56);(1,2)(3,6)(4,5)(7,8)(9,11,13,17)(10,15,14,18)(12,19,20,23)(16,22,21,24)(25,28,30,35)(26,31,27,33)(29,36,37,40)(32,38,34,39)(41,45,47,42)(43,50,44,48)(46,53,54,49)(51,56,52,55);(1,3)(2,6)(4,7)(5,8)(9,12,13,20)(10,16,14,21)(11,19,17,23)(15,22,18,24)(25,29)(26,32)(27,34)(28,36)(30,37)(31,38)(33,39)(35,40)(41,46,47,54)(42,49,45,53)(43,51,44,52)(48,55,50,56)Polycyclic presentation:
2^9|a^2=f,b^2=fgh,c^2=gh,d^2=fg,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, 5870606);