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 | #60938 | 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)(19,23)(20,31)(21,29)(22,27)(24,26)(25,35)(28,32)(30,38)(33,44)(34,40)(36,37)(39,42)(41,47)(43,45)(46,48);(1,3,5,11)(2,6,7,9)(4,12,13,10)(8,16,15,14)(17,19,23,18)(20,31,34,40)(21,29,24,26)(22,37,38,42)(25,35,28,32)(27,39,30,36)(33,41,46,43)(44,45,48,47);(17,20,21,35,23,34,24,32)(18,25,26,31,19,28,29,40)(22,33,36,47,38,46,39,45)(27,41,42,44,30,43,37,48);(2,7)(6,9)(8,15)(14,16)(17,21,23,24)(18,26,19,29)(20,32,34,35)(22,36,38,39)(25,40,28,31)(27,42,30,37)(33,45,46,47)(41,48,43,44);(1,4,5,13)(2,8,7,15)(3,10,11,12)(6,14,9,16)(17,22,23,38)(18,27,19,30)(20,33,34,46)(21,36,24,39)(25,41,28,43)(26,42,29,37)(31,44,40,48)(32,45,35,47)Polycyclic presentation:
2^9|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, 6593314);