Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6469956 |
| 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×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1410 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)(3,8)(4,6)(5,7)(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,23)(20,24)(25,26,31,36)(27,37,39,52)(28,41,43,33)(29,35,30,34)(32,46,48,40)(38,54,42,53)(44,50,45,51)(47,55,49,56);(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(25,27)(26,32)(28,42)(29,40)(30,46)(31,39)(33,49)(34,37)(35,52)(36,48)(38,43)(41,47)(44,56)(45,55)(50,54)(51,53);(1,3,5,4)(2,6,7,8)(9,12,13,20)(10,15,16,24)(11,17,18,19)(14,21,22,23)(25,28,31,43)(26,33,36,41)(27,38,39,42)(29,44,30,45)(32,47,48,49)(34,50,35,51)(37,53,52,54)(40,55,46,56);(1,4,5,3)(2,6,7,8)(11,18)(14,22)(17,19)(21,23)(27,39)(28,43)(32,48)(33,41)(37,52)(40,46)(44,45)(50,51);(10,16)(14,22)(15,24)(21,23)(25,29,31,30)(26,34,36,35)(27,40,39,46)(28,44,43,45)(32,37,48,52)(33,50,41,51)(38,55,42,56)(47,53,49,54)Polycyclic presentation:
2^9|a^2=i,c^2=ghi,d^2=gi,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,f^b=fi,d^c=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6387281);