Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6885065 |
| 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 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1235 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)(3,6)(4,5)(7,8)(9,10)(11,14)(12,16)(13,15)(17,21)(18,20)(19,22)(23,24)(25,26,31,44,30,35,29,34)(27,32,45,56,37,47,38,48)(28,39,43,33,41,51,42,50)(36,52,54,46,40,55,53,49);(1,3,7,4)(2,5,8,6)(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24)(25,27)(26,32)(28,40)(29,38)(30,37)(31,45)(33,49)(34,48)(35,47)(36,41)(39,55)(42,54)(43,53)(44,56)(46,50)(51,52);(1,4,7,3)(2,5,8,6)(10,15)(11,17)(12,19)(20,24)(25,28,30,41)(26,33,35,50)(27,36,37,40)(29,43,31,42)(32,46,47,49)(34,51,44,39)(38,54,45,53)(48,55,56,52);(1,5,7,6)(2,3,8,4)(9,12,13,19)(10,16,15,22)(11,18,17,23)(14,20,21,24)(27,37)(28,41)(32,47)(33,50)(38,45)(39,51)(42,43)(48,56);(1,6,7,5)(2,3,8,4)(11,17)(14,21)(18,23)(20,24)(27,37)(32,47)(36,40)(38,45)(46,49)(48,56)(52,55)(53,54)Polycyclic presentation:
2^9|a^2=gh,b^2=f,c^2=fi,d^2=fhi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,d^b=di,e^b=efh,d^c=dfh,g^c=gi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6855467);