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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1309 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)(3,6)(4,5)(7,8)(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,23)(20,24)(25,26,30,35,31,36,28,33)(27,37,38,46,40,50,44,32)(29,34,42,48,43,49,41,47)(39,51,52,56,54,55,53,45);(2,5)(6,8)(9,11)(10,14)(12,19)(13,17)(15,23)(16,21)(18,20)(22,24)(25,27,31,40)(26,32,36,46)(28,38,30,44)(29,39,43,54)(33,37,35,50)(34,45,49,56)(41,52,42,53)(47,51,48,55);(2,5)(6,8)(11,17)(14,21)(18,19)(22,23)(25,28,31,30)(26,33,36,35)(27,38,40,44)(29,42,43,41)(32,37,46,50)(34,48,49,47)(39,53,54,52)(45,55,56,51);(9,12,13,20)(10,15,16,24)(11,18,17,19)(14,22,21,23)(25,29)(26,34)(27,39)(28,41)(30,42)(31,43)(32,45)(33,47)(35,48)(36,49)(37,51)(38,52)(40,54)(44,53)(46,56)(50,55);(1,3,4,7)(2,6,5,8)(10,16)(14,21)(15,24)(22,23)(29,43)(34,49)(39,54)(41,42)(45,56)(47,48)(51,55)(52,53)Polycyclic presentation:
2^9|a^2=fg,b^2=i,c^2=i,d^2=h,e^2=g,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh,f^b=fi,d^c=di,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6393190);