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 | |
1173 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)(27,32)(28,34)(29,35)(30,44)(31,36)(33,41)(37,46)(38,47)(39,54)(40,48)(42,49)(43,50)(45,51)(52,55)(53,56);(1,3,5,8)(2,6,4,7)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,23)(21,24)(25,27,31,40)(26,32,36,48)(28,37,42,52)(29,38,30,39)(33,45,50,56)(34,46,49,55)(35,47,44,54)(41,51,43,53);(1,2,5,4)(3,7,8,6)(25,28)(26,33)(27,37)(29,41)(30,43)(31,42)(32,45)(34,44)(35,49)(36,50)(38,51)(39,53)(40,52)(46,54)(47,55)(48,56);(9,12,13,21)(10,15,16,20)(11,18,19,24)(14,22,17,23)(25,28)(26,34)(27,37)(29,43)(30,41)(31,42)(32,46)(33,44)(35,50)(36,49)(38,53)(39,51)(40,52)(45,54)(47,56)(48,55);(1,4,5,2)(3,6,8,7)(11,19)(14,17)(18,24)(22,23)(25,29,31,30)(26,35,36,44)(27,38,40,39)(28,41,42,43)(32,47,48,54)(33,49,50,34)(37,51,52,53)(45,55,56,46)Polycyclic presentation:
2^9|a^2=fh,b^2=hi,c^2=h,d^2=f,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, 6762866);