Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470884 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1217 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)(9,10,13,16)(11,14,17,21)(12,19,20,15)(18,24,23,22)(25,26)(27,36)(28,39)(29,32)(30,34)(31,45)(33,43)(35,40)(37,51)(38,47)(41,53)(42,54)(44,50)(46,56)(48,55)(49,52);(2,5)(6,8)(9,11)(10,14)(12,18)(13,17)(15,22)(16,21)(19,24)(20,23)(25,27)(26,32)(28,37)(29,30)(31,38)(33,46)(34,36)(35,47)(39,53)(40,44)(41,42)(43,52)(45,50)(48,49)(51,54)(55,56);(11,17)(14,21)(18,23)(22,24)(25,28,40,55,30,42,31,43)(26,33,45,54,34,48,35,39)(27,37,44,56,29,41,38,52)(32,46,50,51,36,49,47,53);(1,3,4,7)(2,6,5,8)(11,17)(14,21)(18,23)(22,24)(25,29,30,27)(26,32,34,36)(28,41,42,37)(31,44,40,38)(33,46,48,49)(35,47,45,50)(39,53,54,51)(43,56,55,52);(9,12,13,20)(10,15,16,19)(11,18,17,23)(14,22,21,24)(27,29)(32,36)(37,41)(38,44)(46,49)(47,50)(51,53)(52,56)Polycyclic presentation:
2^9|a^2=h,c^2=gi,d^2=f,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dhi,e^b=ei,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6405077);