Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7200042 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1335 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)(12,15,21,24)(18,23,20,22)(25,26,30,32)(27,37,38,48)(28,42,43,34)(29,36,31,35)(33,40,46,41)(39,53,52,51)(44,49,45,50)(47,54,56,55);(1,3)(2,5)(4,7)(6,8)(9,11,13,19)(10,14,16,17)(12,20,21,18)(15,23,24,22)(26,32)(27,38)(28,43)(35,36)(40,41)(44,45)(47,56)(51,53);(3,7)(5,8)(25,27,30,38)(26,33,32,46)(28,39,43,52)(29,41,31,40)(34,47,42,56)(35,37,36,48)(44,55,45,54)(49,53,50,51);(2,6)(3,7)(9,12,13,21)(10,15,16,24)(11,18,19,20)(14,22,17,23)(25,28,30,43)(26,34,32,42)(27,39,38,52)(29,45,31,44)(33,47,46,56)(35,50,36,49)(37,51,48,53)(40,55,41,54);(3,7)(5,8)(10,16)(14,17)(15,24)(22,23)(25,29,30,31)(26,35,32,36)(27,40,38,41)(28,44,43,45)(33,48,46,37)(34,49,42,50)(39,54,52,55)(47,51,56,53)Polycyclic presentation:
2^9|a^2=f,b^2=fi,c^2=i,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ef,g^a=gi,c^b=ch,d^b=dfhi,e^b=ehi,e^c=ei,g^c=gi,e^d=ei59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7110711);