Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6885065 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2283 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1389 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)(10,14)(12,19)(15,21)(17,20)(25,26)(27,36)(28,40)(29,35)(30,34)(31,44)(32,38)(33,42)(37,52)(39,48)(41,49)(43,47)(45,50)(46,53)(51,55)(54,56);(1,3,4,8)(2,5,6,7)(9,10,13,14)(11,15,18,21)(12,16,19,22)(17,23,20,24)(25,27)(26,32)(28,37)(29,43)(30,38)(31,39)(33,46)(34,36)(35,48)(40,56)(41,54)(42,55)(44,47)(45,53)(49,52)(50,51);(3,8)(5,7)(9,11,13,18)(10,15,14,21)(12,20,19,17)(16,24,22,23)(25,28,30,41)(26,33,34,50)(27,37,38,54)(29,42,31,45)(32,46,36,51)(35,49,44,40)(39,53,43,55)(47,56,48,52);(2,6)(5,7)(9,12,13,19)(10,16,14,22)(11,17,18,20)(15,23,21,24)(26,34)(27,38)(28,41)(35,44)(39,43)(42,45)(46,51)(52,56);(10,14)(15,21)(16,22)(23,24)(25,29,30,31)(26,35,34,44)(27,39,38,43)(28,42,41,45)(32,47,36,48)(33,49,50,40)(37,53,54,55)(46,56,51,52)Polycyclic presentation:
2^9|b^2=fi,c^2=fh,d^2=fhi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=dfh,g^c=gi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6861232);