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 | #61329 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1579 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,8)(3,9,11,6)(4,12,13,7)(10,14,16,15)(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)(2,6)(4,10)(5,11)(7,14)(8,9)(12,15)(13,16)(17,19)(18,24)(20,35)(21,30)(22,28)(23,32)(25,43)(26,42)(27,31)(29,36)(33,40)(34,41)(37,48)(38,46)(39,47)(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);(1,4,5,13)(2,7,8,12)(3,10,11,16)(6,14,9,15)(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);(3,11)(6,9)(10,16)(14,15)(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=f,c^2=g,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, 6591357);