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, 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 | |
1174 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,5,4)(3,6,8,7)(9,10,13,16)(11,17,19,14)(12,20,21,15)(18,22,24,23)(25,26,30,36)(27,38,41,32)(28,43,46,34)(29,37,31,35)(33,47,53,45)(39,49,44,54)(40,50,42,51)(48,55,52,56);(1,3,5,8)(2,6,4,7)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,23)(21,24)(25,27,30,41)(26,32,36,38)(28,44,46,39)(29,40,31,42)(33,52,53,48)(34,54,43,49)(35,50,37,51)(45,56,47,55);(1,2,5,4)(3,7,8,6)(25,28)(26,33)(27,39)(29,45)(30,46)(31,47)(32,48)(34,37)(35,43)(36,53)(38,52)(40,55)(41,44)(42,56)(49,51)(50,54);(9,12,13,21)(10,15,16,20)(11,18,19,24)(14,22,17,23)(25,28)(26,34)(27,39)(29,47)(30,46)(31,45)(32,49)(33,37)(35,53)(36,43)(38,54)(40,56)(41,44)(42,55)(48,51)(50,52);(1,4,5,2)(3,6,8,7)(11,19)(14,17)(18,24)(22,23)(25,29,30,31)(26,35,36,37)(27,40,41,42)(28,45,46,47)(32,50,38,51)(33,43,53,34)(39,55,44,56)(48,54,52,49)Polycyclic presentation:
2^9|a^2=fhi,b^2=h,c^2=hi,d^2=fi,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,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, 6763039);