Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682658 |
| 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 | #60709 | 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)(17,18,23,22)(19,28,32,25)(20,33,37,44)(21,39,24,27)(26,35,36,38)(29,46,47,43)(30,31,40,42)(34,41,48,45);(1,3,5,11)(2,6,7,9)(4,12,13,10)(8,16,15,14)(17,19,23,32)(18,25,22,28)(20,34,37,48)(21,30,24,40)(26,43,36,46)(27,31,39,42)(29,35,47,38)(33,45,44,41);(17,20,24,35,23,37,21,38)(18,26,39,44,22,36,27,33)(19,29,40,34,32,47,30,48)(25,41,42,46,28,45,31,43);(2,7)(6,9)(8,15)(14,16)(17,21,23,24)(18,27,22,39)(19,30,32,40)(20,35,37,38)(25,31,28,42)(26,44,36,33)(29,34,47,48)(41,46,45,43);(1,4,5,13)(2,8,7,15)(3,10,11,12)(6,14,9,16)(17,22,23,18)(19,31,32,42)(20,36,37,26)(21,39,24,27)(25,30,28,40)(29,43,47,46)(33,38,44,35)(34,41,48,45)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=gi,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,e^b=efh,h^b=hi,d^c=di,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, 6593317);