Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470428 |
| 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 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1385 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)(4,5)(6,8)(9,10)(11,14)(12,19)(13,15)(16,20)(17,21)(18,24)(22,23)(25,26,30,32)(27,35,39,48)(28,33,40,46)(29,42,36,34)(31,43,44,38)(37,49,51,56)(41,53,50,47)(45,54,55,52);(2,5)(6,7)(9,11,13,17)(10,14,15,21)(12,18,20,23)(16,22,19,24)(25,27,36,43,30,39,29,38)(26,31,42,48,32,44,34,35)(28,37,50,54,40,51,41,52)(33,45,53,56,46,55,47,49);(1,3,4,8)(2,6,5,7)(10,15)(11,17)(16,19)(18,23)(26,32)(31,44)(33,46)(34,42)(35,48)(45,55)(47,53)(49,56);(11,17)(14,21)(18,23)(22,24)(25,28)(26,33)(27,37)(29,41)(30,40)(31,45)(32,46)(34,47)(35,49)(36,50)(38,52)(39,51)(42,53)(43,54)(44,55)(48,56);(9,12,13,20)(10,16,15,19)(11,18,17,23)(14,22,21,24)(28,40)(33,46)(37,51)(41,50)(45,55)(47,53)(49,56)(52,54)Polycyclic presentation:
2^9|a^2=i,b^2=fg,c^2=ghi,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6406628);