Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470003 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1346 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)(25,26,31,35)(27,37,41,52)(28,33,29,34)(30,44,46,36)(32,45,49,40)(38,53,39,54)(42,51,43,50)(47,56,48,55);(9,11)(10,14)(12,19)(13,17)(15,23)(16,21)(18,20)(22,24)(25,27,31,41)(26,32,35,49)(28,39,29,38)(30,45,46,40)(33,48,34,47)(36,52,44,37)(42,55,43,56)(50,53,51,54);(1,3,5,4)(2,6,7,8)(11,17)(14,21)(18,19)(22,23)(25,28)(26,33)(27,38)(29,31)(30,42)(32,47)(34,35)(36,50)(37,53)(39,41)(40,55)(43,46)(44,51)(45,56)(48,49)(52,54);(9,12,13,20)(10,15,16,24)(11,18,17,19)(14,22,21,23)(25,29,31,28)(26,34,35,33)(27,39,41,38)(30,43,46,42)(32,48,49,47)(36,51,44,50)(37,54,52,53)(40,56,45,55);(1,4,5,3)(2,6,7,8)(10,16)(14,21)(15,24)(22,23)(26,35)(28,29)(32,49)(36,44)(37,52)(38,39)(42,43)(55,56)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=g,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,d^c=di,e^c=ei,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6393973);