Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667932 |
| 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 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1223 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(9,10,13,16)(11,17,19,14)(12,15,20,22)(18,23,24,21)(25,26,31,33,29,35,30,36)(27,37,45,47,38,32,40,49)(28,41,46,52,43,34,44,50)(39,48,54,56,42,51,53,55);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23)(25,27)(26,32)(28,42)(29,38)(30,40)(31,45)(33,49)(34,51)(35,37)(36,47)(39,43)(41,48)(44,54)(46,53)(50,55)(52,56);(2,6)(7,8)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(26,33)(27,38)(30,31)(32,47)(34,50)(35,36)(37,49)(39,42)(41,52)(44,46)(48,55)(51,56);(1,3,5,4)(2,7,6,8)(10,16)(14,17)(15,22)(21,23)(25,28)(26,34)(27,39)(29,43)(30,44)(31,46)(32,48)(33,50)(35,41)(36,52)(37,51)(38,42)(40,53)(45,54)(47,55)(49,56);(1,4,5,3)(2,7,6,8)(11,19)(14,17)(18,24)(21,23)(26,35)(28,43)(32,37)(33,36)(39,42)(44,46)(47,49)(53,54)Polycyclic presentation:
2^9|a^2=fg,c^2=fi,d^2=hi,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,d^b=di,e^b=efh,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6776300);