Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519008 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1374 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,6)(3,7,8,5)(9,10,13,12)(11,14,16,15)(17,18,23,26)(19,29,32,27)(20,33,35,25)(21,24,38,41)(22,39,30,28)(31,43,34,45)(36,42,48,40)(37,44,46,47);(1,3,4,8)(2,5,6,7)(9,11)(10,14)(12,15)(13,16)(17,19,30,38,23,32,22,21)(18,24,39,27,26,41,28,29)(20,34,46,36,35,31,37,48)(25,42,47,45,33,40,44,43);(2,6)(3,8)(9,12,13,10)(11,15,16,14)(17,20,23,35)(18,25,26,33)(19,31,32,34)(21,36,38,48)(22,37,30,46)(24,40,41,42)(27,43,29,45)(28,44,39,47);(2,6)(3,8)(11,16)(14,15)(18,26)(19,32)(20,35)(21,38)(28,39)(37,46)(40,42)(43,45);(2,6)(5,7)(17,21,22,32,23,38,30,19)(18,27,28,24,26,29,39,41)(20,36,37,34,35,48,46,31)(25,43,44,40,33,45,47,42)Polycyclic presentation:
2^9|a^2=h,b^2=fi,c^2=ghi,e^2=fgi,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,d^c=di,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7514598);