Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667932 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1242 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(9,10)(11,19)(12,16)(13,18)(14,17)(15,21)(20,24)(22,23)(25,26,31,33,29,35,30,36)(27,37,43,45,38,32,40,47)(28,34,44,48,41,49,42,50)(39,51,54,55,52,46,53,56);(1,2,4,5)(3,7,8,6)(9,11)(10,15)(12,20)(13,22)(14,21)(16,23)(17,19)(18,24)(25,27)(26,32)(28,39)(29,38)(30,40)(31,43)(33,47)(34,46)(35,37)(36,45)(41,52)(42,53)(44,54)(48,56)(49,51)(50,55);(2,5)(6,7)(9,12,14,13)(10,16,17,18)(11,20,21,22)(15,23,19,24)(26,33)(27,38)(30,31)(32,45)(34,48)(35,36)(37,47)(39,52)(42,44)(46,55)(49,50)(51,56);(10,17)(15,19)(16,18)(23,24)(25,28)(26,34)(27,39)(29,41)(30,42)(31,44)(32,46)(33,48)(35,49)(36,50)(37,51)(38,52)(40,53)(43,54)(45,55)(47,56);(1,3,4,8)(2,6,5,7)(9,13,14,12)(10,18,17,16)(11,20,21,22)(15,23,19,24)(28,41)(34,49)(39,52)(42,44)(46,51)(48,50)(53,54)(55,56)Polycyclic presentation:
2^9|a^2=gi,b^2=hi,c^2=fi,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,e^b=efh,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6710200);