Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682638 |
| 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 | |
1628 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)(17,18,23,27)(19,29,32,24)(20,33,30,40)(21,28,34,25)(22,26,38,44)(31,46,47,41)(35,43,37,48)(36,45,39,42);(1,3,5,11)(2,6,7,9)(4,12,13,10)(8,16,15,14)(17,19,23,32)(18,24,27,29)(20,34,30,21)(22,37,38,35)(25,33,28,40)(26,43,44,48)(31,39,47,36)(41,42,46,45);(17,20,34,19,23,30,21,32)(18,24,28,40,27,29,25,33)(22,35,39,31,38,37,36,47)(26,41,45,48,44,46,42,43);(2,7)(6,9)(8,15)(14,16)(17,21,23,34)(18,25,27,28)(19,30,32,20)(22,36,38,39)(24,40,29,33)(26,42,44,45)(31,37,47,35)(41,48,46,43);(1,4,5,13)(2,8,7,15)(3,10,11,12)(6,14,9,16)(17,22,23,38)(18,26,27,44)(19,31,32,47)(20,35,30,37)(21,36,34,39)(24,41,29,46)(25,42,28,45)(33,43,40,48)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,d^b=di,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, 6593327);