Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667980 |
| 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 | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1312 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)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,26,31,45,30,35,29,34)(27,32,37,47,38,48,39,49)(28,40,42,33,44,51,43,52)(36,53,55,46,41,56,54,50);(1,2,4,5)(3,7,8,6)(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23)(25,27)(26,32)(28,41)(29,39)(30,38)(31,37)(33,50)(34,49)(35,48)(36,44)(40,56)(42,54)(43,55)(45,47)(46,52)(51,53);(2,5)(6,7)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(25,28,30,44)(26,33,35,52)(27,36,38,41)(29,42,31,43)(32,46,48,50)(34,51,45,40)(37,54,39,55)(47,53,49,56);(10,16)(14,17)(15,22)(21,23)(25,29,30,31)(26,34,35,45)(27,37,38,39)(28,42,44,43)(32,47,48,49)(33,51,52,40)(36,54,41,55)(46,53,50,56);(1,3,4,8)(2,6,5,7)(11,19)(14,17)(18,24)(21,23)(27,38)(32,48)(36,41)(37,39)(46,50)(47,49)(53,56)(54,55)Polycyclic presentation:
2^9|a^2=gi,b^2=hi,c^2=fi,d^2=i,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,d^b=di,e^b=efh,g^c=gi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6709027);