Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470355 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1434 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2)(3,7)(4,5)(6,8)(9,10)(11,14)(12,19)(13,15)(16,20)(17,21)(18,24)(22,23)(25,26)(27,35)(28,40)(29,44)(30,33)(31,39)(32,42)(34,36)(37,53)(38,50)(41,49)(43,48)(45,56)(46,51)(47,54)(52,55);(2,5)(6,7)(9,11,13,17)(10,14,15,21)(12,18,20,23)(16,22,19,24)(25,27,36,51,30,38,29,39)(26,31,44,50,33,46,34,35)(28,41,52,54,42,37,43,56)(32,47,55,49,40,45,48,53);(1,3,4,8)(2,6,5,7)(10,15)(11,17)(16,19)(18,23)(25,28)(26,32)(27,37)(29,43)(30,42)(31,45)(33,40)(34,48)(35,49)(36,52)(38,41)(39,54)(44,55)(46,47)(50,53)(51,56);(11,17)(14,21)(18,23)(22,24)(27,38)(28,42)(31,46)(32,40)(35,50)(39,51)(43,52)(48,55);(9,12,13,20)(10,16,15,19)(11,18,17,23)(14,22,21,24)(26,33)(31,46)(32,40)(34,44)(35,50)(45,47)(48,55)(49,53)Polycyclic presentation:
2^9|b^2=fg,c^2=gh,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,d^c=di80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6406591);