Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7309060 |
| 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 ◁ #2399 ◁ 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:
(1,2,4,5)(3,6,7,8)(10,14)(11,17)(12,20)(22,24)(25,26,31,33,29,35,30,36)(27,37,43,47,39,32,40,45)(28,34,44,48,41,49,42,50)(38,51,54,56,52,46,53,55);(2,5)(3,7)(9,10,13,14)(11,18,17,15)(12,21,20,16)(19,22,23,24)(25,27,29,39)(26,32,35,37)(28,38,41,52)(30,40,31,43)(33,45,36,47)(34,46,49,51)(42,53,44,54)(48,55,50,56);(9,11)(10,15)(12,19)(13,17)(14,18)(16,22)(20,23)(21,24)(26,33)(30,31)(32,45)(34,48)(35,36)(37,47)(40,43)(42,44)(46,55)(49,50)(51,56)(53,54);(1,3,4,7)(2,6,5,8)(11,17)(15,18)(19,23)(22,24)(25,28)(26,34)(27,38)(29,41)(30,42)(31,44)(32,46)(33,48)(35,49)(36,50)(37,51)(39,52)(40,53)(43,54)(45,55)(47,56);(2,5)(6,8)(9,12,13,20)(10,16,14,21)(11,19,17,23)(15,22,18,24)(26,35)(28,41)(32,37)(33,36)(38,52)(42,44)(45,47)(53,54)Polycyclic presentation:
2^9|a^2=fg,b^2=hi,d^2=fhi,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dfhi,e^b=eh,d^c=dh,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7405861);