Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6885129 |
| 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 | |
1226 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)(25,26,31,43,30,34,29,35)(27,32,44,54,37,46,38,47)(28,39,42,33,41,49,40,48)(36,50,53,45,51,56,52,55);(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)(28,36)(29,38)(30,37)(31,44)(33,45)(34,46)(35,47)(39,50)(40,52)(41,51)(42,53)(43,54)(48,55)(49,56);(1,4,7,3)(2,5,8,6)(10,15)(11,17)(12,19)(20,24)(25,28,30,41)(26,33,34,48)(27,36,37,51)(29,42,31,40)(32,45,46,55)(35,49,43,39)(38,53,44,52)(47,56,54,50);(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)(32,46)(36,51)(38,44)(45,55)(47,54)(50,56)(52,53);(1,6,7,5)(2,3,8,4)(11,17)(14,21)(18,23)(20,24)(26,34)(32,46)(33,48)(35,43)(39,49)(45,55)(47,54)(50,56)Polycyclic presentation:
2^9|a^2=ghi,b^2=f,c^2=fi,d^2=fh,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,d^b=di,e^b=efh,d^c=dfh,g^c=gi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6855545);