Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7309060 |
| 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 | #53074 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1424 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)(11,17)(14,18)(19,23)(22,24)(25,26,31,45,30,36,29,35)(27,32,38,48,40,50,39,49)(28,34,42,33,44,52,43,53)(37,47,56,46,41,54,55,51);(2,5)(3,7)(9,10)(11,18)(12,20)(13,15)(14,17)(16,21)(19,22)(23,24)(25,27,30,40)(26,32,36,50)(28,41,44,37)(29,39,31,38)(33,51,53,46)(34,54,52,47)(35,49,45,48)(42,55,43,56);(9,11)(10,14)(12,19)(13,17)(15,18)(16,22)(20,24)(21,23)(25,28,30,44)(26,33,36,53)(27,37,40,41)(29,42,31,43)(32,46,50,51)(34,35,52,45)(38,55,39,56)(47,49,54,48);(1,3,4,7)(2,6,5,8)(10,15)(11,17)(16,20)(19,23)(25,28)(26,34)(27,37)(29,43)(30,44)(31,42)(32,47)(33,45)(35,53)(36,52)(38,56)(39,55)(40,41)(46,48)(49,51)(50,54);(2,5)(6,8)(9,12,13,21)(10,16,15,20)(11,19,17,23)(14,22,18,24)(25,29,30,31)(26,35,36,45)(27,38,40,39)(28,42,44,43)(32,48,50,49)(33,52,53,34)(37,55,41,56)(46,47,51,54)Polycyclic presentation:
2^9|a^2=fgh,b^2=i,c^2=i,d^2=f,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dfh,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, 7377997);