Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806815 |
| 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 | #525 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1559 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)(19,26)(20,33)(21,28)(22,27)(23,38)(24,29)(25,31)(30,41)(32,40)(34,47)(35,42)(36,48)(37,44)(39,45)(43,46);(1,3,5,11)(2,6,7,9)(4,12,13,10)(8,16,15,14)(17,19)(18,25)(20,24)(21,34)(22,31)(23,32)(26,27)(28,43)(29,40)(30,36)(33,38)(35,46)(37,45)(39,48)(41,44)(42,47);(17,20,23,31,22,32,24,19)(18,26,38,33,27,25,29,40)(21,34,36,45,35,46,37,30)(28,41,48,47,42,39,44,43);(2,7)(6,9)(8,15)(14,16)(18,27)(25,26)(28,42)(29,38)(33,40)(39,41)(43,47)(44,48);(1,4,5,13)(2,8,7,15)(3,10,11,12)(6,14,9,16)(17,21,24,37,22,35,23,36)(18,28,29,44,27,42,38,48)(19,30,32,46,31,45,20,34)(25,39,33,47,26,41,40,43)Polycyclic presentation:
2^9|b^2=fi,c^2=hi,e^2=fhi,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=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6716865);