Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993875 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #60926 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1589 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(4,12)(7,13)(11,16)(14,15)(17,18,23,28,22,27,21,29)(19,30,39,43,33,42,34,24)(20,26,36,25,38,46,37,44)(31,41,32,47,35,40,48,45);(1,2,5,8)(3,9,10,6)(4,13,12,7)(11,14,16,15)(17,19,23,39,22,33,21,34)(18,24,29,42,27,43,28,30)(20,35,37,32,38,31,36,48)(25,41,46,47,44,40,26,45);(1,3)(2,6)(4,11)(5,10)(7,14)(8,9)(12,16)(13,15)(17,20,22,38)(18,25,27,44)(19,31,33,35)(21,36,23,37)(24,40,43,41)(26,29,46,28)(30,47,42,45)(32,39,48,34);(3,10)(6,9)(11,16)(14,15)(17,20)(18,26)(19,32)(21,37)(22,38)(23,36)(24,41)(25,28)(27,46)(29,44)(30,47)(31,34)(33,48)(35,39)(40,43)(42,45);(1,4,5,12)(2,7,8,13)(3,11,10,16)(6,14,9,15)(18,27)(19,33)(25,44)(26,46)(28,29)(31,35)(32,48)(34,39)Polycyclic presentation:
2^9|a^2=gi,b^2=fh,c^2=i,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,f^d=fi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7089608);