Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806839 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2402 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1144 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,4,5)(3,6,7,8)(9,10,14,17)(11,18,21,15)(12,16,13,22)(19,24,20,23)(25,26,32,38)(27,33,40,47)(28,34,44,51)(29,35,31,37)(30,36,45,52)(39,46,53,55)(41,48,54,56)(42,49,43,50);(9,11)(10,15)(12,19)(13,20)(14,21)(16,23)(17,18)(22,24)(25,27,32,40)(26,33,38,47)(28,41,44,54)(29,30,31,45)(34,48,51,56)(35,36,37,52)(39,42,53,43)(46,49,55,50);(2,5)(6,8)(9,12,14,13)(10,16,17,22)(11,19,21,20)(15,23,18,24)(25,28,31,43,32,44,29,42)(26,34,37,50,38,51,35,49)(27,39,30,41,40,53,45,54)(33,46,36,48,47,55,52,56);(9,13,14,12)(10,16,17,22)(11,20,21,19)(15,23,18,24)(25,29,32,31)(26,35,38,37)(27,30,40,45)(28,42,44,43)(33,36,47,52)(34,49,51,50)(39,41,53,54)(46,48,55,56);(1,3,4,7)(2,6,5,8)(11,21)(15,18)(19,20)(23,24)(25,30,31,40,32,45,29,27)(26,36,37,47,38,52,35,33)(28,41,43,53,44,54,42,39)(34,48,50,55,51,56,49,46)Polycyclic presentation:
2^9|a^2=fgi,b^2=i,c^2=fh,d^2=fi,e^2=gh,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6741259);