Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7309080 |
| 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 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1148 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 56 points:
(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,26,31,36,32,37,30,45)(27,33,40,50,41,51,46,49)(28,35,42,52,29,43,44,34)(38,48,54,56,39,55,53,47);(2,6)(3,4)(9,11,13,19)(10,14,16,17)(12,20,21,18)(15,23,24,22)(25,27)(26,33)(28,38)(29,39)(30,46)(31,40)(32,41)(34,47)(35,48)(36,50)(37,51)(42,54)(43,55)(44,53)(45,49)(52,56);(1,2)(3,8)(4,7)(5,6)(25,28,32,29)(26,34,37,52)(27,38,41,39)(30,42,31,44)(33,47,51,56)(35,36,43,45)(40,53,46,54)(48,50,55,49);(1,3,5,4)(2,7,6,8)(9,12,13,21)(10,15,16,24)(11,18,19,20)(14,22,17,23)(25,29,32,28)(26,35,37,43)(27,39,41,38)(30,42,31,44)(33,48,51,55)(34,36,52,45)(40,53,46,54)(47,50,56,49);(1,4,5,3)(2,7,6,8)(10,16)(14,17)(15,24)(22,23)(25,30,32,31)(26,36,37,45)(27,40,41,46)(28,42,29,44)(33,49,51,50)(34,43,52,35)(38,53,39,54)(47,48,56,55)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=i,d^2=fhi,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=efi,c^b=ch,d^b=dfh,e^b=ehi,d^c=dh,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, 7397946);