Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470185 |
| 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 | C22×D4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2295 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1326 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)(25,26,31,33,30,35,29,36)(27,37,43,32,40,50,39,46)(28,34,44,47,41,49,42,48)(38,51,54,45,52,56,53,55);(2,5)(6,8)(9,11)(10,14)(12,19)(13,17)(15,23)(16,21)(18,20)(22,24)(25,27,30,40)(26,32,35,46)(28,38,41,52)(29,43,31,39)(33,37,36,50)(34,45,49,55)(42,54,44,53)(47,51,48,56);(11,17)(14,21)(18,19)(22,23)(26,33)(29,31)(32,37)(34,47)(35,36)(39,43)(42,44)(45,51)(46,50)(48,49)(53,54)(55,56);(9,12,13,20)(10,15,16,24)(11,18,17,19)(14,22,21,23)(25,28)(26,34)(27,38)(29,42)(30,41)(31,44)(32,45)(33,47)(35,49)(36,48)(37,51)(39,53)(40,52)(43,54)(46,55)(50,56);(1,3,4,7)(2,6,5,8)(10,16)(14,21)(15,24)(22,23)(26,35)(28,41)(32,46)(33,36)(37,50)(38,52)(42,44)(53,54)Polycyclic presentation:
2^9|a^2=gi,b^2=i,d^2=h,e^2=fgi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ehi,c^b=ch,d^b=dh,f^b=fi,g^b=gi,f^c=fi,g^c=gi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6393576);