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 | #2407 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1409 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)(25,26,30,47,31,37,32,36)(27,38,40,50,41,33,42,54)(28,43,46,34,29,35,44,51)(39,52,56,53,45,49,55,48);(2,5)(3,7)(9,10)(11,18)(12,20)(13,15)(14,17)(16,21)(19,22)(23,24)(25,27)(26,33)(28,39)(29,45)(30,40)(31,41)(32,42)(34,48)(35,52)(36,50)(37,38)(43,49)(44,55)(46,56)(47,54)(51,53);(9,11)(10,14)(12,19)(13,17)(15,18)(16,22)(20,24)(21,23)(25,28,31,29)(26,34,37,51)(27,39,41,45)(30,44,32,46)(33,48,38,53)(35,47,43,36)(40,55,42,56)(49,50,52,54);(1,3,4,7)(2,6,5,8)(10,15)(11,17)(16,20)(19,23)(25,29,31,28)(26,35,37,43)(27,39,41,45)(30,44,32,46)(33,49,38,52)(34,47,51,36)(40,56,42,55)(48,50,53,54);(2,5)(6,8)(9,12,13,21)(10,16,15,20)(11,19,17,23)(14,22,18,24)(25,30,31,32)(26,36,37,47)(27,40,41,42)(28,44,29,46)(33,50,38,54)(34,35,51,43)(39,55,45,56)(48,52,53,49)Polycyclic presentation:
2^9|a^2=fgh,c^2=i,d^2=f,e^2=hi,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, 7378008);