Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996375 |
| 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 | C2×#553 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1369 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)(3,7)(4,6)(5,8)(9,10,13,16)(11,14,18,22)(12,15,20,24)(17,21,19,23)(25,26,30,35,31,36,29,32)(27,37,40,46,38,33,45,48)(28,34,41,51,44,52,43,47)(39,50,54,55,42,49,53,56);(1,3)(2,5)(4,8)(6,7)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(26,32)(27,38)(28,41)(29,30)(33,48)(35,36)(37,46)(39,53)(42,54)(43,44)(47,51)(49,50);(2,6)(5,7)(9,12,13,20)(10,15,16,24)(11,17,18,19)(14,21,22,23)(25,27)(26,33)(28,42)(29,45)(30,40)(31,38)(32,46)(34,50)(35,48)(36,37)(39,44)(41,53)(43,54)(47,56)(49,52)(51,55);(3,8)(5,7)(12,20)(15,24)(17,19)(21,23)(25,28,30,41,31,44,29,43)(26,34,35,51,36,52,32,47)(27,39,40,54,38,42,45,53)(33,49,48,56,37,50,46,55);(10,16)(11,18)(15,24)(17,19)(25,29,31,30)(26,32,36,35)(27,40,38,45)(28,43,44,41)(33,48,37,46)(34,47,52,51)(39,54,42,53)(49,56,50,55)Polycyclic presentation:
2^9|a^2=fgh,c^2=hi,d^2=fgi,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=ehi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7018182);