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 | #60934 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1637 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)(17,18,21,25)(19,28,31,29)(20,26,34,41)(22,36,23,27)(24,37,39,32)(30,43,46,44)(33,35,42,38)(40,48,45,47);(1,3,5,11)(2,6,7,9)(4,12,13,10)(8,16,15,14)(17,18)(19,29)(20,33)(21,25)(22,27)(23,36)(24,32)(26,38)(28,31)(30,45)(34,42)(35,41)(37,39)(40,46)(43,48)(44,47);(17,19,23,37,21,31,22,32)(18,24,36,29,25,39,27,28)(20,30,38,48,34,46,35,47)(26,40,42,44,41,45,33,43);(2,7)(6,9)(8,15)(14,16)(18,25)(24,39)(26,41)(27,36)(28,29)(33,42)(40,45)(43,44);(1,4,5,13)(2,8,7,15)(3,10,11,12)(6,14,9,16)(17,20,21,34)(18,26,25,41)(19,30,31,46)(22,35,23,38)(24,40,39,45)(27,33,36,42)(28,43,29,44)(32,47,37,48)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=gi,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, 6593303);