Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535664 |
| 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 | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1321 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)(4,5)(7,8)(9,10,13,16)(11,14,17,20)(12,15,19,22)(18,21,23,24)(25,26,30,34)(27,35,38,51)(28,40,42,32)(29,44,36,33)(31,45,47,39)(37,53,41,50)(43,49,52,55)(46,54,48,56);(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24)(25,27,36,45,30,38,29,39)(26,31,44,51,34,47,33,35)(28,41,52,54,42,37,43,56)(32,48,55,50,40,46,49,53);(2,5)(6,8)(11,17)(14,20)(18,23)(21,24)(25,28)(26,32)(27,37)(29,43)(30,42)(31,46)(33,49)(34,40)(35,50)(36,52)(38,41)(39,54)(44,55)(45,56)(47,48)(51,53);(1,3,4,7)(2,6,5,8)(9,12)(10,15)(11,18)(13,19)(14,21)(16,22)(17,23)(20,24);(12,19)(15,22)(18,23)(21,24)(27,38)(28,42)(31,47)(32,40)(35,51)(39,45)(43,52)(49,55)Polycyclic presentation:
2^9|a^2=h,b^2=fi,d^2=gi,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=ei,e^c=ei,e^d=ehi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6472617);