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 | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×D4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2295 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1233 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,5,4)(3,6,8,7)(9,10)(11,19)(12,16)(13,17)(14,18)(15,21)(20,24)(22,23)(25,26)(27,36)(28,40)(29,34)(30,31)(32,39)(33,43)(35,45)(37,48)(38,54)(41,52)(42,51)(44,50)(46,53)(47,55)(49,56);(1,3,5,8)(2,6,4,7)(9,11)(10,15)(12,20)(13,22)(14,21)(16,23)(17,24)(18,19)(26,31)(27,37)(33,40)(34,35)(38,41)(39,53)(50,51)(55,56);(1,2,5,4)(3,7,8,6)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24)(25,27,29,39,30,37,45,53)(26,32,35,36,31,46,34,48)(28,41,44,55,43,38,42,56)(33,49,50,54,40,47,51,52);(9,12,14,13)(10,17,18,16)(11,20,21,22)(15,24,19,23)(25,28)(26,33)(27,38)(29,44)(30,43)(31,40)(32,47)(34,51)(35,50)(36,52)(37,41)(39,56)(42,45)(46,49)(48,54)(53,55);(1,4,5,2)(3,6,8,7)(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(25,29,30,45)(26,34,31,35)(27,39,37,53)(28,42,43,44)(32,48,46,36)(33,50,40,51)(38,55,41,56)(47,52,49,54)Polycyclic presentation:
2^9|a^2=hi,b^2=hi,c^2=fgh,d^2=fi,e^2=fhi,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^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6765700);