Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667964 |
| 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 | #61182 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1676 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18)(19,31)(20,35)(21,30)(22,28)(23,29)(24,27)(25,33)(26,37)(32,43)(34,45)(36,42)(38,44)(39,47)(40,48)(41,46);(1,3,5,11)(2,6,8,9)(4,10,12,16)(7,13,14,15)(17,19,23,33)(18,25,29,31)(20,36,38,40)(21,32,24,34)(22,39,41,37)(26,46,47,28)(27,43,30,45)(35,48,44,42);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,20,23,38)(18,26,29,47)(19,22,33,41)(21,37,24,39)(25,42,31,48)(27,35,30,44)(28,43,46,45)(32,40,34,36);(2,8)(6,9)(7,14)(13,15)(17,21,23,24)(18,27,29,30)(19,32,33,34)(20,37,38,39)(22,40,41,36)(25,43,31,45)(26,35,47,44)(28,42,46,48);(3,11)(6,9)(10,16)(13,15)(17,22,23,41)(18,28,29,46)(19,20,33,38)(21,36,24,40)(25,44,31,35)(26,43,47,45)(27,48,30,42)(32,39,34,37)Polycyclic presentation:
2^9|b^2=f,c^2=f,d^2=i,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,g^c=gi,h^c=hi,e^d=ei,h^e=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6577107);