Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994139 |
| 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 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1318 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)(3,5)(4,6)(7,8)(9,10,13,16)(11,14,17,21)(12,19,20,15)(18,24,23,22)(25,26,31,36)(27,37,38,53)(28,33,41,54)(29,34,40,55)(30,45,47,35)(32,48,51,39)(42,56,43,50)(44,49,46,52);(1,3,4,7)(2,5,6,8)(11,17)(12,20)(14,21)(15,19)(25,27,30,39,31,38,47,48)(26,32,35,37,36,51,45,53)(28,40,42,46,41,29,43,44)(33,52,50,55,54,49,56,34);(2,6)(5,8)(9,11)(10,14)(12,18)(13,17)(15,22)(16,21)(19,24)(20,23)(25,28)(26,33)(27,29)(30,42)(31,41)(32,49)(34,37)(35,50)(36,54)(38,40)(39,44)(43,47)(45,56)(46,48)(51,52)(53,55);(11,17)(14,21)(18,23)(22,24)(25,29,31,40)(26,34,36,55)(27,28,38,41)(30,46,47,44)(32,50,51,56)(33,53,54,37)(35,49,45,52)(39,43,48,42);(9,12,13,20)(10,15,16,19)(11,18,17,23)(14,22,21,24)(27,38)(28,41)(32,51)(33,54)(37,53)(39,48)(42,43)(50,56)Polycyclic presentation:
2^9|a^2=h,b^2=fg,d^2=i,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fi,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,e^c=ei,f^d=fi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7080402);