Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682440 |
| 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 ◁ #2295 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1175 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 56 points:
(9,10,14,17)(11,19,21,15)(12,16,13,18)(20,24,22,23)(25,26)(27,36)(28,40)(29,35)(30,32)(31,46)(33,39)(34,43)(37,49)(38,54)(41,53)(42,52)(44,47)(45,51)(48,55)(50,56);(1,2)(3,6)(4,5)(7,8)(9,11,14,21)(10,15,17,19)(12,20,13,22)(16,23,18,24)(26,32)(27,37)(34,40)(35,46)(38,41)(39,44)(51,52)(55,56);(2,5)(6,8)(9,12,14,13)(10,16,17,18)(11,20,21,22)(15,23,19,24)(25,27,31,39,30,37,29,44)(26,33,35,36,32,47,46,49)(28,41,42,55,43,38,45,56)(34,50,52,54,40,48,51,53);(1,3,4,7)(2,6,5,8)(10,17)(15,19)(16,18)(23,24)(25,28,30,43)(26,34,32,40)(27,38,37,41)(29,45,31,42)(33,48,47,50)(35,52,46,51)(36,53,49,54)(39,56,44,55);(2,5)(6,8)(9,13,14,12)(10,18,17,16)(11,20,21,22)(15,23,19,24)(25,29,30,31)(26,35,32,46)(27,39,37,44)(28,42,43,45)(33,49,47,36)(34,51,40,52)(38,55,41,56)(48,53,50,54)Polycyclic presentation:
2^9|a^2=fi,b^2=fi,c^2=fgi,d^2=h,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=di,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6696316);