Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6469995 |
| 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×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1482 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)(3,5)(4,6)(7,8)(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,23)(20,24)(25,26,31,36)(27,37,38,52)(28,33,43,50)(29,35,30,34)(32,45,46,40)(39,53,41,56)(42,51,44,49)(47,54,48,55);(1,3,4,7)(2,5,6,8)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(25,27)(26,32)(28,41)(29,40)(30,45)(31,38)(33,48)(34,37)(35,52)(36,46)(39,43)(42,55)(44,54)(47,50)(49,53)(51,56);(2,6)(5,8)(9,12,13,20)(10,15,16,24)(11,17,18,19)(14,21,22,23)(27,38)(32,46)(37,52)(39,41)(40,45)(47,48)(53,56)(54,55);(11,18)(14,22)(17,19)(21,23)(25,28)(26,33)(27,39)(29,44)(30,42)(31,43)(32,47)(34,51)(35,49)(36,50)(37,53)(38,41)(40,55)(45,54)(46,48)(52,56);(10,16)(14,22)(15,24)(21,23)(25,29,31,30)(26,34,36,35)(27,40,38,45)(28,42,43,44)(32,37,46,52)(33,49,50,51)(39,54,41,55)(47,56,48,53)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=hi,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,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6386778);