Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6907993 |
| 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 | C2×#121 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1398 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)(3,7)(4,6)(5,8)(9,10,13,16)(11,17,19,14)(12,20,21,15)(18,22,24,23)(25,26)(27,32)(28,38)(29,34)(30,44)(31,35)(33,40)(36,48)(37,45)(39,52)(41,47)(42,49)(43,50)(46,53)(51,56)(54,55);(1,3)(2,5)(4,8)(6,7)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,23)(21,24)(25,27,31,30)(26,32,35,44)(28,36,41,40)(29,42,43,37)(33,38,48,47)(34,49,50,45)(39,53,54,51)(46,55,56,52);(3,8)(5,7)(12,21)(15,20)(18,24)(22,23)(25,28,27,36,31,41,30,40)(26,33,44,47,35,48,32,38)(29,39,37,51,43,54,42,53)(34,46,49,55,50,56,45,52);(2,6)(5,7)(9,12,13,21)(10,15,16,20)(11,18,19,24)(14,22,17,23)(25,29,31,43)(26,34,35,50)(27,37,30,42)(28,39,41,54)(32,45,44,49)(33,46,48,56)(36,51,40,53)(38,52,47,55);(11,19)(14,17)(18,24)(22,23)(25,27,31,30)(26,32,35,44)(28,40,41,36)(29,42,43,37)(33,47,48,38)(34,49,50,45)(39,51,54,53)(46,52,56,55)Polycyclic presentation:
2^9|a^2=fh,b^2=i,c^2=gi,d^2=fhi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=dfh,e^c=ei,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6855844);