Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682648 |
| 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 | #60935 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1618 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)(4,8)(5,7)(6,11)(10,16)(12,14)(13,15)(17,18)(19,29)(20,25)(21,26)(22,27)(23,28)(24,31)(30,33)(32,43)(34,35)(36,42)(37,40)(38,41)(39,45)(44,47)(46,48);(1,3,5,11)(2,6,7,9)(4,12,13,10)(8,16,15,14)(17,18)(19,30)(20,27)(21,35)(22,25)(23,28)(24,31)(26,34)(29,33)(32,39)(36,41)(37,40)(38,42)(43,45)(44,46)(47,48);(17,19,20,31,23,33,22,37)(18,24,27,29,28,40,25,30)(21,32,34,45,36,47,38,46)(26,39,41,43,42,48,35,44);(2,7)(6,9)(8,15)(14,16)(17,20,23,22)(18,25,28,27)(19,31,33,37)(21,34,36,38)(24,30,40,29)(26,35,42,41)(32,45,47,46)(39,44,48,43);(1,4,5,13)(2,8,7,15)(3,10,11,12)(6,14,9,16)(17,21)(18,26)(19,32)(20,34)(22,38)(23,36)(24,39)(25,35)(27,41)(28,42)(29,43)(30,44)(31,45)(33,47)(37,46)(40,48)Polycyclic presentation:
2^9|b^2=f,c^2=g,d^2=i,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,h^b=hi,h^c=hi,h^e=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6593304);