Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438991 |
| 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 | C2×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1817 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)(3,7)(4,8)(5,6)(9,10,13,15)(11,14,12,16)(17,18,22,27,23,28,36,37)(19,29,20,25,32,39,31,24)(21,26,34,42,35,43,44,47)(30,41,33,40,46,48,45,38);(2,6)(7,8)(9,11,13,12)(10,14,15,16)(17,19,23,32)(18,24,28,25)(20,22,31,36)(21,33,35,45)(26,41,43,48)(27,39,37,29)(30,44,46,34)(38,47,40,42);(1,3,5,4)(2,7,6,8)(9,12,13,11)(10,14,15,16)(17,20,22,32,23,31,36,19)(18,25,27,39,28,24,37,29)(21,33,34,46,35,45,44,30)(26,40,42,48,43,38,47,41);(1,4,5,3)(2,8,6,7)(17,21,22,34,23,35,36,44)(18,26,27,42,28,43,37,47)(19,30,20,33,32,46,31,45)(24,38,29,41,25,40,39,48);(17,22,23,36)(18,27,28,37)(19,31,32,20)(21,34,35,44)(24,39,25,29)(26,42,43,47)(30,45,46,33)(38,48,40,41)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,c^2=fg,d^2=f,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^c=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6365769);