Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6469995 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1356 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)(3,8)(4,6)(5,7)(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,23)(20,24)(25,26)(27,36)(28,34)(29,35)(30,43)(31,33)(32,38)(37,51)(39,47)(40,49)(41,48)(42,50)(44,45)(46,53)(52,55)(54,56);(9,11)(10,14)(12,19)(13,17)(15,23)(16,21)(18,20)(22,24)(25,27,31,39)(26,32,33,45)(28,37,41,54)(29,38,30,44)(34,46,48,55)(35,47,43,36)(40,52,42,53)(49,51,50,56);(1,3,5,4)(2,6,7,8)(11,17)(14,21)(18,19)(22,23)(26,33)(32,45)(34,48)(35,43)(36,47)(46,55)(49,50)(51,56);(9,12,13,20)(10,15,16,24)(11,18,17,19)(14,22,21,23)(25,28)(26,34)(27,37)(29,42)(30,40)(31,41)(32,46)(33,48)(35,50)(36,51)(38,53)(39,54)(43,49)(44,52)(45,55)(47,56);(1,4,5,3)(2,6,7,8)(10,16)(14,21)(15,24)(22,23)(25,29,31,30)(26,35,33,43)(27,38,39,44)(28,40,41,42)(32,47,45,36)(34,49,48,50)(37,52,54,53)(46,51,55,56)Polycyclic presentation:
2^9|b^2=i,c^2=gi,d^2=h,e^2=g,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,f^b=fi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6393947);