Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4465744 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1496 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,5)(3,7)(10,14)(12,16)(17,18,21,24)(19,22,25,28)(20,23,27,30)(26,29,31,32)(34,35)(39,41)(40,42)(46,47);(2,5)(6,8)(10,14)(11,15)(19,25)(22,28)(26,31)(29,32)(33,34,38,35)(36,41,44,39)(37,42,45,40)(43,46,48,47);(1,2)(3,6)(4,5)(7,8)(9,10,13,14)(11,12,15,16)(17,18)(19,22)(20,23)(21,24)(25,28)(26,29)(27,30)(31,32)(33,35,38,34)(36,39,44,41)(37,42,45,40)(43,46,48,47);(1,2)(3,6)(4,5)(7,8)(9,11,13,15)(10,12,14,16)(17,19,21,25)(18,22,24,28)(20,26,27,31)(23,29,30,32)(33,36,38,44)(34,39,35,41)(37,43,45,48)(40,46,42,47);(1,3)(2,6)(4,7)(5,8)(9,12)(10,15)(11,14)(13,16)(17,20,21,27)(18,23,24,30)(19,26,25,31)(22,29,28,32)(33,37,38,45)(34,40,35,42)(36,43,44,48)(39,46,41,47)Polycyclic presentation:
2^9|a^2=fgh,b^2=f,c^2=fhi,d^2=gi,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4433965);