Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470185 |
| 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 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1348 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)(27,37,41,45)(28,34,42,49)(29,35,30,33)(32,40,47,39)(38,51,52,55)(43,50,44,48)(46,54,56,53);(1,3,5,4)(2,6,7,8)(9,11)(10,14)(12,19)(13,17)(15,23)(16,21)(18,20)(22,24)(25,27,31,41)(26,32,36,47)(28,38,42,52)(29,40,30,39)(33,45,35,37)(34,46,49,56)(43,54,44,53)(48,55,50,51);(1,4,5,3)(2,6,7,8)(11,17)(14,21)(18,19)(22,23)(26,33)(29,30)(32,45)(34,48)(35,36)(37,47)(39,40)(43,44)(46,55)(49,50)(51,56)(53,54);(9,12,13,20)(10,15,16,24)(11,18,17,19)(14,22,21,23)(25,28,31,42)(26,34,36,49)(27,38,41,52)(29,43,30,44)(32,46,47,56)(33,48,35,50)(37,51,45,55)(39,53,40,54);(10,16)(14,21)(15,24)(22,23)(28,42)(34,49)(38,52)(43,44)(46,56)(48,50)(51,55)(53,54)Polycyclic presentation:
2^9|a^2=i,b^2=fgi,c^2=fg,d^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,f^b=fi,g^b=gi,f^c=fi,g^c=gi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6392250);