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 | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1266 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)(4,8)(5,6)(9,10)(11,14)(12,19)(13,15)(16,20)(17,21)(18,24)(22,23)(25,26,30,33)(27,31,36,46)(28,38,42,55)(29,43,40,34)(32,47,49,41)(35,50,39,54)(37,53,51,45)(44,56,48,52);(2,6)(7,8)(9,11)(10,14)(12,18)(13,17)(15,21)(16,22)(19,24)(20,23)(25,27,30,36)(26,31,33,46)(28,39,42,35)(29,37,40,51)(32,48,49,44)(34,45,43,53)(38,54,55,50)(41,56,47,52);(1,3,5,4)(2,7,6,8)(10,15)(11,17)(16,19)(18,23)(25,28,40,47,30,42,29,41)(26,32,43,55,33,49,34,38)(27,35,51,56,36,39,37,52)(31,44,53,54,46,48,45,50);(1,4,5,3)(2,8,6,7)(11,17)(14,21)(18,23)(22,24)(25,27)(26,31)(28,35)(29,37)(30,36)(32,44)(33,46)(34,45)(38,50)(39,42)(40,51)(41,52)(43,53)(47,56)(48,49)(54,55);(9,12,13,20)(10,16,15,19)(11,18,17,23)(14,22,21,24)(26,33)(27,36)(32,49)(34,43)(35,39)(37,51)(38,55)(52,56)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fgh,d^2=f,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,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, 6406278);