Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6433057 |
| 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 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1337 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)(11,13)(12,14)(15,16)(17,18,24,30)(19,26,33,39)(20,25,32,40)(21,27,35,42)(22,29,23,28)(31,41,45,48)(34,38,46,47)(36,44,37,43);(1,3,5,4)(2,6,8,7)(9,11)(10,13)(12,15)(14,16)(17,19,23,32,24,33,22,20)(18,25,29,39,30,40,28,26)(21,31,37,46,35,45,36,34)(27,38,44,48,42,47,43,41);(1,3,5,4)(2,7,8,6)(11,15)(13,16)(17,20,22,33,24,32,23,19)(18,26,28,40,30,39,29,25)(21,34,36,45,35,46,37,31)(27,41,43,47,42,48,44,38);(10,14)(13,16)(17,21,24,35)(18,27,30,42)(19,31,33,45)(20,34,32,46)(22,36,23,37)(25,38,40,47)(26,41,39,48)(28,43,29,44);(1,4,5,3)(2,6,8,7)(21,35)(27,42)(31,45)(34,46)(36,37)(38,47)(41,48)(43,44)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=f,d^2=i,e^2=fg,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,g^a=gi,c^b=ch,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6341854);