Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470149 |
| 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 | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2280 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1428 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,23)(20,24)(25,26)(27,32)(28,39)(29,34)(30,43)(31,35)(33,40)(36,51)(37,46)(38,47)(41,49)(42,48)(44,50)(45,54)(52,55)(53,56);(1,3,4,8)(2,5,6,7)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(25,27,31,38)(26,32,35,47)(28,36,41,53)(29,42,30,37)(33,45,50,55)(34,48,43,46)(39,51,49,56)(40,54,44,52);(9,12,13,20)(10,15,16,24)(11,17,18,19)(14,21,22,23)(25,28)(26,33)(27,36)(29,40)(30,44)(31,41)(32,45)(34,49)(35,50)(37,52)(38,53)(39,43)(42,54)(46,51)(47,55)(48,56);(11,18)(14,22)(17,19)(21,23)(25,29,31,30)(26,34,35,43)(27,37,38,42)(28,40,41,44)(32,46,47,48)(33,49,50,39)(36,52,53,54)(45,51,55,56);(10,16)(14,22)(15,24)(21,23)(27,38)(32,47)(36,53)(37,42)(45,55)(46,48)(51,56)(52,54)Polycyclic presentation:
2^9|a^2=f,b^2=fi,c^2=h,d^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dhi,e^b=ei,g^c=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6385012);