Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #6132643 |
| automorphism | Aut(G) order 1048576 |
| 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 | |
1292 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,6)(3,7,8,5)(9,10,13,16)(11,14,15,12)(17,18)(19,23)(20,28)(21,22)(24,29)(25,26)(27,32)(30,31)(34,35)(36,43)(37,45)(46,47);(3,8)(5,7)(11,15)(12,14)(18,22)(19,25)(24,28)(27,31)(33,34,38,35)(36,39,43,41)(37,42,45,40)(44,47,48,46);(9,11,13,15)(10,14,16,12)(17,18)(19,26)(20,28)(21,22)(23,25)(24,29)(27,30)(31,32)(33,35,38,34)(36,41,43,39)(37,40,45,42)(44,46,48,47);(1,3)(2,5)(4,8)(6,7)(9,12,13,14)(10,15,16,11)(17,19,21,25)(18,23,22,26)(20,27,29,31)(24,30,28,32)(33,36)(34,39)(35,41)(37,44)(38,43)(40,46)(42,47)(45,48);(1,2)(3,5)(4,6)(7,8)(17,20,21,29)(18,24,22,28)(19,27,25,31)(23,30,26,32)(33,37,38,45)(34,40,35,42)(36,44,43,48)(39,46,41,47)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=fh,d^2=h,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dfi,e^b=eg,d^c=dfg,e^c=ef62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6080853);