Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| 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 | |
1374 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)(5,8)(11,15)(13,16)(17,18,21,22)(19,25,27,23)(20,28,30,24)(26,31,29,32)(33,34,38,41)(35,36,42,39)(37,45,46,40)(43,47,48,44);(9,10,12,14)(11,13,15,16)(18,22)(23,25)(24,28)(31,32)(35,42)(36,39)(43,48)(44,47);(1,2,4,6)(3,5,7,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,23)(20,28)(21,22)(24,30)(25,27)(26,32)(29,31)(33,35)(34,39)(36,41)(37,43)(38,42)(40,47)(44,45)(46,48);(1,3,4,7)(2,5,6,8)(9,11)(10,13)(12,15)(14,16)(17,19)(18,23)(20,29)(21,27)(22,25)(24,32)(26,30)(28,31)(33,36,38,39)(34,35,41,42)(37,44,46,47)(40,43,45,48);(17,20,21,30)(18,24,22,28)(19,26,27,29)(23,31,25,32)(33,37)(34,40)(35,43)(36,44)(38,46)(39,47)(41,45)(42,48)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=hi,d^2=fghi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3444705);