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 | |
1308 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)(9,10)(11,15)(12,16)(13,17)(14,18)(19,22)(20,23)(21,24)(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);(9,11,14,21)(10,15,18,24)(12,20,13,19)(16,23,17,22)(25,27,30,37)(26,31,33,45)(28,39,40,36)(29,42,43,38)(32,46,48,44)(34,50,49,35)(41,52,55,54)(47,51,56,53);(1,3,5,4)(2,6,7,8)(9,12,14,13)(10,16,18,17)(11,19,21,20)(15,22,24,23)(25,28,30,40)(26,32,33,48)(27,36,37,39)(29,41,43,55)(31,44,45,46)(34,47,49,56)(35,51,50,53)(38,52,42,54);(9,13,14,12)(10,17,18,16)(11,20,21,19)(15,23,24,22)(27,37)(31,45)(35,50)(36,39)(38,42)(44,46)(51,53)(52,54);(1,4,5,3)(2,6,7,8)(10,18)(15,24)(16,17)(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=h,c^2=gh,d^2=hi,e^2=g,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, 6397900);