Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6632088 |
| 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 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1386 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)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,26,32,36,31,37,30,47)(27,38,43,54,42,33,41,50)(28,35,44,51,29,45,46,34)(39,53,55,48,40,49,56,52);(1,3)(2,5)(4,7)(6,8)(9,11,13,19)(10,14,16,17)(12,18,20,24)(15,21,22,23)(25,27,31,42)(26,33,37,38)(28,39,29,40)(30,41,32,43)(34,48,51,52)(35,49,45,53)(36,50,47,54)(44,55,46,56);(3,7)(5,8)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(25,28,31,29)(26,34,37,51)(27,39,42,40)(30,44,32,46)(33,48,38,52)(35,36,45,47)(41,55,43,56)(49,50,53,54);(3,7)(5,8)(10,16)(11,19)(15,22)(18,24)(25,29,31,28)(26,35,37,45)(27,40,42,39)(30,44,32,46)(33,49,38,53)(34,36,51,47)(41,55,43,56)(48,50,52,54);(2,6)(5,8)(10,16)(14,17)(15,22)(21,23)(25,30,31,32)(26,36,37,47)(27,41,42,43)(28,44,29,46)(33,50,38,54)(34,45,51,35)(39,55,40,56)(48,53,52,49)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=f,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,c^b=ch,d^b=dfhi,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, 6542090);