Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533123 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1308 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)(9,10,13,16)(11,17,18,14)(12,15,20,22)(19,23,24,21)(25,26,31,35,30,34,29,36)(27,32,38,49,40,46,39,47)(28,41,42,33,44,50,43,48)(37,51,54,45,53,56,52,55);(2,5)(6,8)(9,11)(10,14)(12,19)(13,18)(15,21)(16,17)(20,24)(22,23)(25,27)(26,32)(28,37)(29,39)(30,40)(31,38)(33,45)(34,46)(35,49)(36,47)(41,51)(42,54)(43,52)(44,53)(48,55)(50,56);(11,18)(14,17)(19,24)(21,23)(25,28,30,44)(26,33,34,48)(27,37,40,53)(29,42,31,43)(32,45,46,55)(35,41,36,50)(38,52,39,54)(47,56,49,51);(1,3,4,7)(2,6,5,8)(12,20)(15,22)(19,24)(21,23)(26,34)(32,46)(33,48)(35,36)(41,50)(45,55)(47,49)(51,56);(9,12,13,20)(10,15,16,22)(11,19,18,24)(14,21,17,23)(25,29,30,31)(26,35,34,36)(27,38,40,39)(28,42,44,43)(32,47,46,49)(33,41,48,50)(37,52,53,54)(45,56,55,51)Polycyclic presentation:
2^9|a^2=fgi,c^2=i,d^2=fh,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,c^b=ch,e^b=ei,g^c=gi,e^d=eh71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6451742);