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 | |
1394 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(3,7)(6,8)(9,10,13,12)(11,14,16,15)(17,18,21,24)(19,25,26,22)(20,28,29,23)(27,30,32,31)(33,34,37,38)(35,41,42,39)(36,44,45,40)(43,46,48,47);(2,5)(6,8)(9,11,13,16)(10,14,12,15)(19,26)(22,25)(27,32)(30,31)(34,38)(35,42)(36,45)(46,47);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,15)(13,16)(17,18)(19,22)(20,23)(21,24)(25,26)(27,30)(28,29)(31,32)(33,35,37,42)(34,39,38,41)(36,43,45,48)(40,46,44,47);(1,3,4,7)(2,6,5,8)(9,12,13,10)(11,15,16,14)(17,19,21,26)(18,22,24,25)(20,27,29,32)(23,30,28,31)(33,34)(35,41)(36,40)(37,38)(39,42)(43,47)(44,45)(46,48);(17,20,21,29)(18,23,24,28)(19,27,26,32)(22,30,25,31)(33,36,37,45)(34,40,38,44)(35,43,42,48)(39,46,41,47)Polycyclic presentation:
2^9|a^2=fhi,b^2=fghi,c^2=f,d^2=gh,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, 1935451);