Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| 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 | C22×#41 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1428 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)(3,5)(4,6)(7,8)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,26,29,33,30,34,40,46)(27,31,37,43,35,41,48,52)(28,32,39,45,38,44,50,54)(36,42,49,53,47,51,55,56);(1,3,4,7)(2,5,6,8)(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23)(25,27)(26,31)(28,36)(29,37)(30,35)(32,42)(33,43)(34,41)(38,47)(39,49)(40,48)(44,51)(45,53)(46,52)(50,55)(54,56);(2,6)(5,8)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(27,35)(28,38)(31,41)(32,44)(37,48)(39,50)(43,52)(45,54);(11,19)(14,17)(18,24)(21,23)(25,28)(26,32)(27,36)(29,39)(30,38)(31,42)(33,45)(34,44)(35,47)(37,49)(40,50)(41,51)(43,53)(46,54)(48,55)(52,56);(10,16)(14,17)(15,22)(21,23)(28,38)(32,44)(36,47)(39,50)(42,51)(45,54)(49,55)(53,56)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=f,h^2=i,b^a=bf,c^a=cg,e^a=ef,c^b=ci,d^b=df,d^c=di,e^d=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6293758);