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 | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×D4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2295 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1197 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
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)(27,37)(28,41)(29,35)(30,36)(31,46)(32,40)(33,42)(34,43)(38,48)(39,50)(44,49)(45,51)(47,54)(52,56)(53,55);(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)(28,38,43,55)(29,44,31,39)(33,47,51,52)(34,48,41,53)(35,50,46,49)(42,56,45,54);(1,2,5,4)(3,7,8,6)(25,28)(26,33)(27,38)(29,42)(30,43)(31,45)(32,47)(34,35)(36,51)(37,52)(39,54)(40,55)(41,46)(44,56)(48,50)(49,53);(9,12,13,21)(10,15,16,20)(11,18,19,24)(14,22,17,23)(25,28)(26,34)(27,38)(29,45)(30,43)(31,42)(32,48)(33,35)(36,41)(37,53)(39,56)(40,55)(44,54)(46,51)(47,50)(49,52);(1,4,5,2)(3,6,8,7)(11,19)(14,17)(18,24)(22,23)(25,29,30,31)(26,35,36,46)(27,39,40,44)(28,42,43,45)(32,49,37,50)(33,34,51,41)(38,54,55,56)(47,53,52,48)Polycyclic presentation:
2^9|a^2=fhi,b^2=hi,c^2=h,d^2=fi,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,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, 6762884);