Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470884 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2295 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1175 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,5,8)(3,7,4,6)(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,23)(20,24)(25,26,32,36)(27,34,41,50)(28,33,40,49)(29,35,45,56)(30,38,31,37)(39,51,43,53)(42,48,44,52)(46,55,47,54);(1,3,5,4)(2,6,8,7)(9,11)(10,14)(12,19)(13,17)(15,23)(16,21)(18,20)(22,24)(25,27,30,28,32,41,31,40)(26,33,37,34,36,49,38,50)(29,43,46,44,45,39,47,42)(35,52,54,53,56,48,55,51);(1,3,5,4)(2,7,8,6)(11,17)(14,21)(18,19)(22,23)(25,28,30,41,32,40,31,27)(26,34,37,49,36,50,38,33)(29,44,46,39,45,42,47,43)(35,53,54,48,56,51,55,52);(9,12,13,20)(10,15,16,24)(11,18,17,19)(14,22,21,23)(25,29)(26,35)(27,39)(28,42)(30,46)(31,47)(32,45)(33,48)(34,51)(36,56)(37,54)(38,55)(40,44)(41,43)(49,52)(50,53);(1,4,5,3)(2,6,8,7)(10,16)(14,21)(15,24)(22,23)(26,36)(29,45)(33,49)(34,50)(37,38)(39,43)(42,44)(46,47)Polycyclic presentation:
2^9|a^2=fgi,b^2=f,c^2=f,d^2=hi,e^2=fg,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,d^c=di,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6393766);