Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | C24 | C23×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1366 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)(3,7)(4,6)(5,8)(9,10)(11,18)(12,16)(13,22)(14,17)(15,21)(19,24)(20,23)(25,26,29,33,30,34,40,46)(27,31,37,43,36,42,49,53)(28,32,39,45,38,44,50,54)(35,41,48,52,47,51,55,56);(3,8)(5,7)(9,11,14,21)(10,15,17,18)(12,20,13,19)(16,24,22,23)(25,27)(26,31)(28,35)(29,37)(30,36)(32,41)(33,43)(34,42)(38,47)(39,48)(40,49)(44,51)(45,52)(46,53)(50,55)(54,56);(1,3,4,8)(2,5,6,7)(9,12,14,13)(10,16,17,22)(11,19,21,20)(15,23,18,24)(25,28)(26,32)(27,35)(29,39)(30,38)(31,41)(33,45)(34,44)(36,47)(37,48)(40,50)(42,51)(43,52)(46,54)(49,55)(53,56);(2,6)(5,7)(9,13,14,12)(10,16,17,22)(11,20,21,19)(15,23,18,24)(28,38)(32,44)(35,47)(39,50)(41,51)(45,54)(48,55)(52,56);(27,36)(31,42)(35,47)(37,49)(41,51)(43,53)(48,55)(52,56)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=fg,d^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dfg,c^b=cfg,d^b=df,e^b=ei,d^c=di92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6314844);