Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7309080 |
| 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 | #2407 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1401 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)(25,26,30,36,31,37,32,38)(27,39,42,51,43,33,44,50)(28,45,47,34,29,35,46,52)(40,49,56,53,41,54,55,48);(2,5)(3,7)(9,10)(11,18)(12,20)(13,15)(14,17)(16,21)(19,22)(23,24)(25,27,31,43)(26,33,37,39)(28,40,29,41)(30,42,32,44)(34,48,52,53)(35,49,45,54)(36,50,38,51)(46,55,47,56);(9,11)(10,14)(12,19)(13,17)(15,18)(16,22)(20,24)(21,23)(25,28,31,29)(26,34,37,52)(27,40,43,41)(30,46,32,47)(33,48,39,53)(35,36,45,38)(42,55,44,56)(49,50,54,51);(1,3,4,7)(2,6,5,8)(10,15)(11,17)(16,20)(19,23)(25,29,31,28)(26,35,37,45)(27,41,43,40)(30,46,32,47)(33,49,39,54)(34,36,52,38)(42,55,44,56)(48,50,53,51);(2,5)(6,8)(9,12,13,21)(10,16,15,20)(11,19,17,23)(14,22,18,24)(25,30,31,32)(26,36,37,38)(27,42,43,44)(28,46,29,47)(33,50,39,51)(34,45,52,35)(40,55,41,56)(48,54,53,49)Polycyclic presentation:
2^9|a^2=fgh,b^2=i,c^2=i,d^2=f,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=efi,c^b=ch,d^b=dfhi,e^b=eh,d^c=dh,g^c=gi,e^d=ei59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7378070);