Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667772 |
| 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×D4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1243 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,5,7)(3,6,4,8)(9,10,13,16)(11,17,19,14)(12,20,21,15)(18,22,24,23)(25,26,31,36)(27,32,40,48)(28,33,43,51)(29,34,30,35)(37,45,54,56)(38,46,39,47)(41,49,53,55)(42,50,44,52);(1,3,5,4)(2,6,7,8)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,23)(21,24)(25,27)(26,32)(28,41)(29,38)(30,39)(31,40)(33,49)(34,46)(35,47)(36,48)(37,42)(43,53)(44,54)(45,50)(51,55)(52,56);(2,7)(6,8)(25,28,31,43)(26,33,36,51)(27,37,40,54)(29,42,30,44)(32,45,48,56)(34,50,35,52)(38,53,39,41)(46,55,47,49);(9,12,13,21)(10,15,16,20)(11,18,19,24)(14,22,17,23)(25,29,31,30)(26,34,36,35)(27,38,40,39)(28,42,43,44)(32,46,48,47)(33,50,51,52)(37,53,54,41)(45,55,56,49);(1,4,5,3)(2,8,7,6)(11,19)(14,17)(18,24)(22,23)(27,39)(29,30)(32,47)(34,35)(37,41)(38,40)(42,44)(45,49)(46,48)(50,52)(53,54)(55,56)Polycyclic presentation:
2^9|a^2=fgi,b^2=g,c^2=i,d^2=fi,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6626742);