Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5956507 |
| 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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1461 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)(18,22)(19,25)(24,29)(27,31)(33,34,38,35)(36,41,43,39)(37,42,45,40)(44,46,48,47);(1,2,4,5)(3,8,7,6)(9,10,13,14)(11,12,15,16)(18,22)(23,26)(24,29)(30,32)(36,43)(39,41)(44,48)(46,47);(3,7)(6,8)(10,14)(11,15)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31)(33,35,38,34)(36,41,43,39)(37,42,45,40)(44,47,48,46);(1,3)(2,6)(4,7)(5,8)(9,11)(10,12)(13,15)(14,16)(17,19,21,25)(18,23,22,26)(20,27,28,31)(24,30,29,32)(33,36,38,43)(34,39,35,41)(37,44,45,48)(40,46,42,47);(1,2)(3,6)(4,5)(7,8)(9,12)(10,11)(13,16)(14,15)(17,20,21,28)(18,24,22,29)(19,27,25,31)(23,30,26,32)(33,37)(34,40)(35,42)(36,44)(38,45)(39,46)(41,47)(43,48)Polycyclic presentation:
2^9|a^2=fhi,b^2=f,c^2=fhi,d^2=fghi,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dgi,e^b=ef,d^c=dfg62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6029328);