Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1396 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(9,10,12,14)(11,13,15,16)(17,18,22,19)(20,23,28,25)(21,24,29,26)(27,30,32,31)(34,35)(36,43)(37,45)(46,47);(2,5)(6,8)(11,15)(13,16)(18,19)(21,29)(23,25)(27,32)(33,34,38,35)(36,41,43,39)(37,42,45,40)(44,46,48,47);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,15)(14,16)(17,19,22,18)(20,23,28,25)(21,26,29,24)(27,30,32,31)(33,35,38,34)(36,39,43,41)(37,42,45,40)(44,46,48,47);(1,3,4,7)(2,6,5,8)(9,10)(11,13)(12,14)(15,16)(17,20)(18,23)(19,25)(21,27)(22,28)(24,30)(26,31)(29,32)(33,36,38,43)(34,39,35,41)(37,44,45,48)(40,46,42,47);(17,21)(18,24)(19,26)(20,27)(22,29)(23,30)(25,31)(28,32)(33,37,38,45)(34,40,35,42)(36,44,43,48)(39,46,41,47)Polycyclic presentation:
2^9|a^2=fi,b^2=g,c^2=f,d^2=fghi,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1751275);