Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667900 |
| 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 | #2535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1374 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,5,8)(3,6,4,7)(9,10,13,16)(11,17,19,14)(12,20,21,15)(18,22,24,23)(25,26)(27,31)(28,33)(29,32)(30,35)(34,42)(36,44)(37,43)(38,46)(39,48)(40,49)(41,47)(45,52)(50,54)(51,55)(53,56);(3,4)(6,7)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,23)(21,24)(25,27,30,38)(26,31,35,46)(28,39,40,36)(29,37,42,52)(32,43,34,45)(33,48,49,44)(41,53,54,51)(47,56,50,55);(1,3,5,4)(2,6,8,7)(26,32)(29,42)(31,43)(33,47)(34,35)(37,52)(41,54)(44,55)(45,46)(48,56)(49,50)(51,53);(9,12,13,21)(10,15,16,20)(11,18,19,24)(14,22,17,23)(25,28)(26,33)(27,36)(29,41)(30,40)(31,44)(32,47)(34,50)(35,49)(37,51)(38,39)(42,54)(43,55)(45,56)(46,48)(52,53);(1,4,5,3)(2,7,8,6)(11,19)(14,17)(18,24)(22,23)(28,40)(33,49)(36,39)(41,54)(44,48)(47,50)(51,53)(55,56)Polycyclic presentation:
2^9|a^2=fh,b^2=i,c^2=h,d^2=f,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,g^c=gi,e^d=ei71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6755129);