Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6469983 |
| 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 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1316 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)(3,6)(4,8)(5,7)(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,23)(20,24)(25,26,29,34,30,33,43,49)(27,35,38,45,37,50,42,31)(28,32,41,47,40,48,55,56)(36,51,52,46,39,53,54,44);(1,3,5,4)(2,6,7,8)(9,11)(10,14)(12,19)(13,17)(15,23)(16,21)(18,20)(22,24)(25,27)(26,31)(28,39)(29,42)(30,37)(32,46)(33,45)(34,50)(35,49)(36,40)(38,43)(41,52)(44,48)(47,51)(53,56)(54,55);(1,4,5,3)(2,6,7,8)(11,17)(14,21)(18,19)(22,23)(25,28)(26,32)(27,36)(29,41)(30,40)(31,44)(33,48)(34,47)(35,51)(37,39)(38,52)(42,54)(43,55)(45,46)(49,56)(50,53);(9,12,13,20)(10,15,16,24)(11,18,17,19)(14,22,21,23)(27,37)(31,45)(35,50)(36,39)(38,42)(44,46)(51,53)(52,54);(10,16)(14,21)(15,24)(22,23)(26,33)(28,40)(31,45)(34,49)(35,50)(36,39)(41,55)(52,54)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=g,d^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh,f^b=fi,e^c=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6392031);