Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6660385 |
| 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×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1455 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)(3,5)(4,6)(7,8)(9,10,13,16)(11,17,19,14)(12,20,21,15)(18,22,24,23)(25,26,29,33)(27,35,37,31)(28,39,40,48)(30,43,42,34)(32,44,47,41)(36,51,52,56)(38,46,54,50)(45,53,49,55);(3,7)(5,8)(9,11,13,19)(10,14,16,17)(12,18,21,24)(15,22,20,23)(25,27)(26,31)(28,36)(29,37)(30,38)(32,45)(33,35)(34,46)(39,56)(40,52)(41,53)(42,54)(43,50)(44,55)(47,49)(48,51);(1,3)(2,5)(4,7)(6,8)(12,21)(15,20)(18,24)(22,23)(25,28,29,40)(26,32,33,47)(27,36,37,52)(30,44,42,41)(31,45,35,49)(34,48,43,39)(38,55,54,53)(46,51,50,56);(3,7)(5,8)(9,12,13,21)(10,15,16,20)(11,18,19,24)(14,22,17,23)(26,33)(28,40)(31,35)(34,43)(36,52)(41,44)(46,50)(53,55);(2,6)(5,8)(10,16)(14,17)(15,20)(22,23)(26,33)(27,37)(32,47)(34,43)(36,52)(38,54)(39,48)(53,55)Polycyclic presentation:
2^9|a^2=f,b^2=fi,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=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6568165);