Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682380 |
| 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×D4 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1323 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)(25,26,31,35)(27,32,37,47)(28,39,43,55)(29,34,30,45)(33,41,51,42)(36,52,40,54)(38,48,44,49)(46,56,50,53);(1,3)(2,5)(4,7)(6,8)(9,11,13,19)(10,14,16,17)(12,18,21,24)(15,22,20,23)(25,27)(26,32)(28,40)(29,44)(30,38)(31,37)(33,50)(34,49)(35,47)(36,43)(39,54)(41,53)(42,56)(45,48)(46,51)(52,55);(2,6)(3,7)(25,28,29,41,31,43,30,42)(26,33,45,55,35,51,34,39)(27,36,44,56,37,40,38,53)(32,46,48,54,47,50,49,52);(9,12,13,21)(10,15,16,20)(11,18,19,24)(14,22,17,23)(27,37)(32,47)(36,40)(38,44)(46,50)(48,49)(52,54)(53,56);(3,7)(5,8)(11,19)(14,17)(18,24)(22,23)(25,29,31,30)(26,34,35,45)(27,38,37,44)(28,41,43,42)(32,48,47,49)(33,39,51,55)(36,53,40,56)(46,54,50,52)Polycyclic presentation:
2^9|a^2=fi,b^2=f,c^2=gh,d^2=f,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=efh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6670531);