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 | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×D4 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1320 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)(25,26,30,35,31,36,29,43)(27,32,39,49,40,50,44,48)(28,34,41,53,42,52,45,33)(37,54,56,51,38,47,55,46);(2,5)(3,7)(9,10)(11,17)(12,20)(13,15)(14,18)(16,21)(19,22)(23,24)(25,27,31,40)(26,32,36,50)(28,38,42,37)(29,44,30,39)(33,51,53,46)(34,47,52,54)(35,49,43,48)(41,55,45,56);(9,11)(10,14)(12,19)(13,18)(15,17)(16,22)(20,23)(21,24)(25,28,31,42)(26,33,36,53)(27,37,40,38)(29,41,30,45)(32,46,50,51)(34,35,52,43)(39,55,44,56)(47,48,54,49);(1,3,4,7)(2,6,5,8)(10,15)(11,18)(16,20)(19,24)(25,28)(26,34)(27,38)(29,45)(30,41)(31,42)(32,47)(33,43)(35,53)(36,52)(37,40)(39,55)(44,56)(46,49)(48,51)(50,54);(2,5)(6,8)(9,12,13,21)(10,16,15,20)(11,19,18,24)(14,22,17,23)(25,29,31,30)(26,35,36,43)(27,39,40,44)(28,41,42,45)(32,48,50,49)(33,52,53,34)(37,55,38,56)(46,54,51,47)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=i,d^2=f,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=efi,c^b=ch,d^b=dfhi,e^b=eh,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, 7367000);