Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7324820 |
| 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 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1381 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(9,10)(11,18)(12,16)(13,22)(14,17)(15,20)(19,24)(21,23)(25,26)(27,35)(28,33)(29,31)(30,42)(32,39)(34,37)(36,48)(38,47)(40,44)(41,53)(43,50)(45,52)(46,51)(49,55)(54,56);(1,2)(3,7)(4,6)(5,8)(9,11,14,20)(10,15,17,18)(12,19,13,21)(16,23,22,24)(26,31)(27,36)(33,44)(34,42)(38,49)(39,50)(46,53)(52,56);(1,3)(2,5)(4,8)(6,7)(9,12,14,13)(10,16,17,22)(11,19,20,21)(15,23,18,24)(25,27,37,50,29,36,30,39)(26,32,42,48,31,43,34,35)(28,38,51,56,40,49,41,52)(33,45,53,55,44,54,46,47);(2,6)(3,8)(11,20)(15,18)(19,21)(23,24)(25,28,29,40)(26,33,31,44)(27,38,36,49)(30,41,37,51)(32,45,43,54)(34,46,42,53)(35,47,48,55)(39,52,50,56);(2,6)(5,7)(9,13,14,12)(10,16,17,22)(11,21,20,19)(15,23,18,24)(26,31)(28,40)(32,43)(34,42)(35,48)(38,49)(41,51)(52,56)Polycyclic presentation:
2^9|b^2=fi,c^2=fg,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=ehi,d^c=dhi,e^d=ei56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7242299);