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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1660 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
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)(19,28,32,24)(20,34,38,44)(21,27,40,26)(25,36,37,39)(29,43,47,48)(30,42,33,31)(35,45,46,41);(1,3,5,11)(2,6,7,9)(4,12,13,10)(8,16,15,14)(17,19,23,32)(18,24,22,28)(20,35,38,46)(21,30,40,33)(25,43,37,48)(26,42,27,31)(29,39,47,36)(34,41,44,45);(17,20,21,39,23,38,40,36)(18,25,26,34,22,37,27,44)(19,29,30,35,32,47,33,46)(24,41,42,48,28,45,31,43);(2,7)(6,9)(8,15)(14,16)(17,21,23,40)(18,26,22,27)(19,30,32,33)(20,36,38,39)(24,42,28,31)(25,44,37,34)(29,46,47,35)(41,43,45,48);(1,4,5,13)(2,8,7,15)(3,10,11,12)(6,14,9,16)(17,22,23,18)(19,31,32,42)(20,37,38,25)(21,27,40,26)(24,33,28,30)(29,43,47,48)(34,39,44,36)(35,41,46,45)Polycyclic presentation:
2^9|a^2=i,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,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, 6593315);