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 | #392 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1244 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)(11,19)(12,16)(13,17)(14,18)(15,21)(20,24)(22,23)(25,26)(27,36)(28,40)(29,44)(30,35)(31,34)(32,39)(33,43)(37,47)(38,48)(41,50)(42,51)(45,49)(46,52)(53,55)(54,56);(9,11,14,21)(10,15,18,19)(12,20,13,22)(16,23,17,24)(25,27)(26,32)(28,41)(29,38)(30,39)(31,45)(33,50)(34,48)(35,36)(37,42)(40,56)(43,54)(44,49)(46,55)(47,52)(51,53);(2,5)(6,8)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24)(25,28)(26,33)(27,37)(29,42)(30,43)(31,46)(32,47)(34,51)(35,40)(36,53)(38,54)(39,55)(41,45)(44,52)(48,50)(49,56);(1,3,4,7)(2,6,5,8)(9,12,14,13)(10,17,18,16)(11,20,21,22)(15,24,19,23)(25,29,30,31)(26,34,35,44)(27,38,39,45)(28,42,43,46)(32,48,36,49)(33,51,40,52)(37,54,55,41)(47,50,53,56);(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(26,35)(27,38)(29,31)(32,48)(33,40)(36,49)(37,54)(39,45)(41,55)(42,46)(47,50)(53,56)Polycyclic presentation:
2^9|b^2=fi,c^2=fi,d^2=fgi,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,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, 6617487);