Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806647 |
| 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 | C22×D4 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1320 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)(3,6)(4,5)(7,8)(9,10)(11,18)(12,16)(13,22)(14,17)(15,21)(19,24)(20,23)(25,26,31,36)(27,32,38,46)(28,33,40,48)(29,34,43,51)(30,35,44,52)(37,45,54,56)(39,47,41,49)(42,50,53,55);(9,11)(10,15)(12,19)(13,20)(14,21)(16,23)(17,18)(22,24)(25,27,31,38)(26,32,36,46)(28,39,40,41)(29,30,43,44)(33,47,48,49)(34,35,51,52)(37,53,54,42)(45,55,56,50);(2,5)(6,8)(9,12,14,13)(10,16,17,22)(11,19,21,20)(15,23,18,24)(25,28,30,42,31,40,44,53)(26,33,35,50,36,48,52,55)(27,37,29,39,38,54,43,41)(32,45,34,47,46,56,51,49);(9,13,14,12)(10,16,17,22)(11,20,21,19)(15,23,18,24)(28,40)(33,48)(37,54)(39,41)(42,53)(45,56)(47,49)(50,55);(1,3,4,7)(2,6,5,8)(11,21)(15,18)(19,20)(23,24)(25,29,30,38,31,43,44,27)(26,34,35,46,36,51,52,32)(28,41,42,37,40,39,53,54)(33,49,50,45,48,47,55,56)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fh,d^2=f,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=di,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6738676);