Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6499823 |
| 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 ◁ #2283 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1303 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)(3,8)(4,6)(5,7)(25,26,29,34)(27,36,37,31)(28,33,41,50)(30,32,43,35)(38,48,40,45)(39,44,51,46)(42,47,54,49)(52,55,53,56);(1,3,5,4)(2,6,7,8)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,27,29,37)(26,31,34,36)(28,40,41,38)(30,39,43,51)(32,46,35,44)(33,48,50,45)(42,53,54,52)(47,55,49,56);(9,11,13,19)(10,14,16,17)(12,20,21,18)(15,23,24,22)(26,32)(27,37)(28,41)(30,43)(31,44)(33,49)(34,35)(36,46)(45,56)(47,50)(48,55)(52,53);(9,12,13,21)(10,15,16,24)(11,18,19,20)(14,22,17,23)(25,28,29,41)(26,33,34,50)(27,38,37,40)(30,42,43,54)(31,45,36,48)(32,47,35,49)(39,52,51,53)(44,55,46,56);(1,4,5,3)(2,6,7,8)(10,16)(14,17)(15,24)(22,23)(27,37)(28,41)(31,36)(33,50)(39,51)(42,54)(44,46)(47,49)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=hi,d^2=h,e^2=fi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh,g^c=gi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6437523);