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 | #61215 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1663 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)(3,10)(4,8)(5,14)(6,9)(7,13)(11,16)(12,15)(17,18,23,21)(19,24,32,39)(20,33,35,44)(22,27,40,26)(25,38,36,37)(28,45,46,42)(29,30,43,31)(34,41,47,48);(1,3)(2,7)(4,11)(5,12)(6,13)(8,15)(9,10)(14,16)(17,19)(18,24)(20,34)(21,39)(22,30)(23,32)(25,42)(26,29)(27,43)(28,38)(31,40)(33,41)(35,47)(36,45)(37,46)(44,48);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,15,10,16)(17,20,22,37,23,35,40,38)(18,25,26,33,21,36,27,44)(19,28,30,34,32,46,31,47)(24,41,29,45,39,48,43,42);(1,5,6,4)(2,8,9,14)(3,12,13,11)(7,15,10,16)(20,35)(25,36)(28,46)(33,44)(34,47)(37,38)(41,48)(42,45);(3,13)(7,10)(11,12)(15,16)(17,21,23,18)(19,29,32,43)(20,36,35,25)(22,27,40,26)(24,30,39,31)(28,45,46,42)(33,37,44,38)(34,48,47,41)Polycyclic presentation:
2^9|a^2=i,c^2=fg,d^2=f,e^2=i,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, 6591761);