Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806839 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #392 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1411 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,14)(12,13)(15,16)(17,18,23,29)(19,30,33,25)(20,34,38,26)(21,40,24,27)(22,28,32,42)(31,41,46,45)(35,43,37,47)(36,44,39,48);(1,3,4,8)(2,5,6,7)(17,19,23,33)(18,25,29,30)(20,35,38,37)(21,32,24,22)(26,43,34,47)(27,42,40,28)(31,36,46,39)(41,48,45,44);(10,13)(14,16)(17,20,24,39,23,38,21,36)(18,26,27,44,29,34,40,48)(19,31,32,35,33,46,22,37)(25,41,28,43,30,45,42,47);(2,6)(5,7)(17,21,23,24)(18,27,29,40)(19,32,33,22)(20,36,38,39)(25,42,30,28)(26,44,34,48)(31,35,46,37)(41,47,45,43);(3,8)(5,7)(9,11,12,15)(10,14,13,16)(17,22,21,33,23,32,24,19)(18,28,40,25,29,42,27,30)(20,37,36,46,38,35,39,31)(26,43,48,41,34,47,44,45)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=h,d^2=i,e^2=gh,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6737178);