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 | |
1175 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)(27,37,41,32)(28,42,44,34)(29,35,31,47)(33,46,52,43)(38,53,39,49)(40,51,45,50)(48,56,54,55);(1,3,5,8)(2,6,4,7)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,23)(21,24)(25,27,30,41)(26,32,36,37)(28,38,44,39)(29,45,31,40)(33,48,52,54)(34,53,42,49)(35,51,47,50)(43,56,46,55);(1,2,5,4)(3,7,8,6)(25,28)(26,33)(27,38)(29,43)(30,44)(31,46)(32,48)(34,35)(36,52)(37,54)(39,41)(40,55)(42,47)(45,56)(49,50)(51,53);(9,12,13,21)(10,15,16,20)(11,18,19,24)(14,22,17,23)(25,28)(26,34)(27,39)(29,46)(30,44)(31,43)(32,49)(33,35)(36,42)(37,53)(38,41)(40,55)(45,56)(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)(27,40,41,45)(28,43,44,46)(32,50,37,51)(33,34,52,42)(38,55,39,56)(48,49,54,53)Polycyclic presentation:
2^9|a^2=fh,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,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, 6762996);