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 | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1275 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)(3,7)(4,8)(5,6)(9,10,13,16)(11,14,17,20)(12,15,19,22)(18,21,23,24)(25,26)(27,36)(28,33)(29,34)(30,45)(31,35)(32,39)(37,51)(38,47)(40,52)(41,55)(42,49)(43,50)(44,56)(46,54)(48,53);(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24)(25,27,30,39,31,40,29,38)(26,32,34,36,35,47,45,52)(28,41,44,53,42,37,43,54)(33,48,50,55,49,46,56,51);(2,6)(7,8)(11,17)(14,20)(18,23)(21,24)(25,28)(26,33)(27,37)(29,43)(30,44)(31,42)(32,46)(34,50)(35,49)(36,51)(38,53)(39,54)(40,41)(45,56)(47,48)(52,55);(1,3,5,4)(2,7,6,8)(9,12)(10,15)(11,18)(13,19)(14,21)(16,22)(17,23)(20,24)(28,42)(33,49)(37,41)(43,44)(46,48)(50,56)(51,55)(53,54);(1,4,5,3)(2,8,6,7)(12,19)(15,22)(18,23)(21,24)(25,29,31,30)(26,34,35,45)(27,38,40,39)(28,43,42,44)(32,36,47,52)(33,50,49,56)(37,53,41,54)(46,51,48,55)Polycyclic presentation:
2^9|a^2=hi,b^2=f,d^2=g,e^2=gi,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^c=di,e^d=ehi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6472812);