Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667900 |
| 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 | |
1179 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)(27,37,40,32)(28,42,45,34)(29,35,31,47)(33,46,52,44)(38,49,43,53)(39,54,41,50)(48,55,51,56);(1,3,5,8)(2,6,4,7)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,23)(21,24)(25,27,30,40)(26,32,36,37)(28,43,45,38)(29,39,31,41)(33,51,52,48)(34,53,42,49)(35,50,47,54)(44,56,46,55);(1,2,5,4)(3,7,8,6)(25,28)(26,33)(27,38)(29,44)(30,45)(31,46)(32,48)(34,35)(36,52)(37,51)(39,55)(40,43)(41,56)(42,47)(49,50)(53,54);(9,12,13,21)(10,15,16,20)(11,18,19,24)(14,22,17,23)(25,28)(26,34)(27,38)(29,46)(30,45)(31,44)(32,49)(33,35)(36,42)(37,53)(39,56)(40,43)(41,55)(47,52)(48,50)(51,54);(1,4,5,2)(3,6,8,7)(11,19)(14,17)(18,24)(22,23)(25,29,30,31)(26,35,36,47)(27,39,40,41)(28,44,45,46)(32,50,37,54)(33,34,52,42)(38,55,43,56)(48,49,51,53)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,g^a=gi,c^b=ch,d^b=di,e^b=efh,g^c=gi,e^d=ei71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6763007);