Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500535 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
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 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1310 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)(9,10)(11,14)(12,16)(13,15)(17,21)(18,20)(19,22)(23,24)(25,26)(27,34)(28,33)(29,35)(30,36)(31,37)(32,38)(39,50)(40,49)(41,48)(42,47)(43,56)(44,52)(45,53)(46,54)(51,55);(2,5)(6,8)(9,11,13,17)(10,14,15,21)(12,18,19,23)(16,20,22,24)(25,27,30,40,32,41,31,28)(26,33,37,48,38,49,36,34)(29,39,45,43,46,55,44,42)(35,47,52,51,54,56,53,50);(1,3,4,7)(2,6,5,8)(10,15)(11,17)(12,19)(20,24)(25,28,31,41,32,40,30,27)(26,34,36,49,38,48,37,33)(29,43,44,39,46,42,45,55)(35,51,53,47,54,50,52,56);(9,12,13,19)(10,16,15,22)(11,18,17,23)(14,20,21,24)(25,29,32,46)(26,35,38,54)(27,39,41,55)(28,42,40,43)(30,45,31,44)(33,47,49,56)(34,50,48,51)(36,53,37,52);(11,17)(14,21)(18,23)(20,24)(25,30,32,31)(26,36,38,37)(27,40,41,28)(29,44,46,45)(33,34,49,48)(35,52,54,53)(39,42,55,43)(47,51,56,50)Polycyclic presentation:
2^9|a^2=fgh,b^2=g,c^2=f,d^2=hi,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=eh,d^c=dhi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6433755);