Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470901 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2295 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1205 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2)(3,8)(4,6)(5,7)(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,23)(20,24)(25,26,32,38)(27,39,44,51)(28,45,46,54)(29,35,40,50)(30,37,31,36)(33,43,52,42)(34,47,53,41)(48,56,49,55);(1,3,5,4)(2,6,7,8)(9,11)(10,14)(12,19)(13,17)(15,23)(16,21)(18,20)(22,24)(25,27,31,43,32,44,30,42)(26,33,37,51,38,52,36,39)(28,40,41,48,46,29,47,49)(34,50,45,55,53,35,54,56);(11,17)(14,21)(18,19)(22,23)(25,28,30,47,32,46,31,41)(26,34,36,54,38,53,37,45)(27,40,42,49,44,29,43,48)(33,50,39,56,52,35,51,55);(9,12,13,20)(10,15,16,24)(11,18,17,19)(14,22,21,23)(25,29,32,40)(26,35,38,50)(27,41,44,47)(28,43,46,42)(30,48,31,49)(33,45,52,54)(34,51,53,39)(36,55,37,56);(1,4,5,3)(2,6,7,8)(10,16)(14,21)(15,24)(22,23)(28,46)(29,40)(34,53)(35,50)(41,47)(45,54)(48,49)(55,56)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=gi,d^2=hi,e^2=fg,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dhi,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6394486);