Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6660434 |
| 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 | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1477 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2)(3,5)(4,6)(7,8)(9,10,13,16)(11,17,19,14)(12,20,21,15)(18,22,24,23)(25,26)(27,32)(28,41)(29,34)(30,45)(31,36)(33,43)(35,38)(37,53)(39,48)(40,51)(42,47)(44,52)(46,50)(49,54)(55,56);(3,7)(5,8)(9,11,13,19)(10,14,16,17)(12,18,21,24)(15,22,20,23)(25,27)(26,32)(28,37)(29,38)(30,40)(31,39)(33,47)(34,35)(36,48)(41,53)(42,43)(44,54)(45,51)(46,55)(49,52)(50,56);(1,3)(2,5)(4,7)(6,8)(12,21)(15,20)(18,24)(22,23)(25,28,31,44)(26,33,36,50)(27,37,39,54)(29,42,40,55)(30,46,38,43)(32,47,48,56)(34,49,51,53)(35,52,45,41);(3,7)(5,8)(9,12,13,21)(10,15,16,20)(11,18,19,24)(14,22,17,23)(25,29,31,40)(26,34,36,51)(27,38,39,30)(28,42,44,55)(32,35,48,45)(33,49,50,53)(37,43,54,46)(41,47,52,56);(2,6)(5,8)(10,16)(14,17)(15,20)(22,23)(27,39)(29,40)(32,48)(34,51)(37,54)(42,55)(47,56)(49,53)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=i,d^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,g^a=gi,c^b=ch,e^b=ei,d^c=dfh,g^c=gi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6568180);