Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806623 |
| 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 | |
1149 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)(25,26,30,34)(27,37,39,32)(28,40,42,33)(29,35,45,47)(31,36,44,51)(38,46,54,52)(41,48,56,49)(43,55,53,50);(9,11)(10,15)(12,19)(13,20)(14,21)(16,23)(17,18)(22,24)(25,27,30,39)(26,32,34,37)(28,41,42,56)(29,31,45,44)(33,48,40,49)(35,51,47,36)(38,53,54,43)(46,55,52,50);(2,5)(6,8)(9,12,14,13)(10,16,17,22)(11,19,21,20)(15,23,18,24)(25,28,31,43,30,42,44,53)(26,33,36,50,34,40,51,55)(27,38,29,41,39,54,45,56)(32,46,47,48,37,52,35,49);(9,13,14,12)(10,16,17,22)(11,20,21,19)(15,23,18,24)(26,34)(28,42)(32,37)(35,47)(36,51)(38,54)(41,56)(43,53);(1,3,4,7)(2,6,5,8)(11,21)(15,18)(19,20)(23,24)(25,29,44,27,30,45,31,39)(26,35,51,37,34,47,36,32)(28,41,53,38,42,56,43,54)(33,49,55,52,40,48,50,46)Polycyclic presentation:
2^9|a^2=fgi,b^2=i,c^2=fhi,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,d^c=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6741184);