Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6469983 |
| 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 | #2619 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1542 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)(3,8)(4,7)(5,6)(9,10)(11,14)(12,15)(13,16)(17,18,21,26,22,25,35,41)(19,27,30,37,29,42,34,23)(20,24,33,39,32,40,47,48)(28,43,44,38,31,45,46,36);(2,6)(7,8)(9,11,14,10)(12,15,13,16)(17,19,22,29)(18,23,25,37)(20,31,32,28)(21,34,35,30)(24,38,40,36)(26,42,41,27)(33,44,47,46)(39,43,48,45);(1,3,5,4)(2,7,6,8)(10,11)(15,16)(17,20,22,32)(18,24,25,40)(19,28,29,31)(21,33,35,47)(23,36,37,38)(26,39,41,48)(27,43,42,45)(30,44,34,46);(9,12,14,13)(10,15,11,16)(19,29)(23,37)(27,42)(28,31)(30,34)(36,38)(43,45)(44,46);(1,4,5,3)(2,7,6,8)(9,13,14,12)(10,15,11,16)(18,25)(20,32)(23,37)(26,41)(27,42)(28,31)(33,47)(44,46)Polycyclic presentation:
2^9|a^2=fg,b^2=h,c^2=gh,d^2=hi,e^2=g,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh,f^b=fi,e^c=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6397976);