Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438943 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2285 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1654 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)(4,8)(5,6)(9,10,12,14)(11,15,16,13)(17,18,23,28,24,30,22,29)(19,31,34,25,36,42,35,43)(20,26,37,44,38,45,21,27)(32,47,33,40,39,41,48,46);(2,6)(7,8)(17,19)(18,25)(20,33)(21,39)(22,34)(23,35)(24,36)(26,41)(27,46)(28,31)(29,42)(30,43)(32,37)(38,48)(40,44)(45,47);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,13,14,15)(17,20,23,37,24,38,22,21)(18,26,28,44,30,45,29,27)(19,32,34,33,36,39,35,48)(25,40,42,41,43,46,31,47);(1,4,5,3)(2,8,6,7)(17,21,22,38,24,37,23,20)(18,27,29,45,30,44,28,26)(19,33,35,32,36,48,34,39)(25,41,31,40,43,47,42,46);(17,22,24,23)(18,28,30,29)(19,34,36,35)(20,21,38,37)(25,31,43,42)(26,44,45,27)(32,33,39,48)(40,47,46,41)Polycyclic presentation:
2^9|a^2=gh,c^2=fgi,d^2=f,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,h^b=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6365536);