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 | |
1371 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,15)(12,13)(14,16)(18,22)(19,25)(24,29)(27,31)(33,34)(35,37)(36,43)(38,46)(39,41)(40,47)(42,48)(44,45);(3,8)(5,7)(9,11)(10,14)(12,15)(13,16)(18,22)(23,26)(24,29)(30,32)(33,35,39,36)(34,37,41,43)(38,42,47,44)(40,45,46,48);(11,16)(14,15)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31)(33,36,39,35)(34,37,41,43)(38,44,47,42)(40,45,46,48);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15)(12,14)(13,16)(17,19,21,25)(18,23,22,26)(20,27,28,31)(24,30,29,32)(33,37)(34,36)(35,41)(38,45)(39,43)(40,44)(42,46)(47,48);(1,2)(3,5)(4,6)(7,8)(9,12,13,10)(11,14,16,15)(17,20,21,28)(18,24,22,29)(19,27,25,31)(23,30,26,32)(33,38,39,47)(34,40,41,46)(35,42,36,44)(37,45,43,48)Polycyclic presentation:
2^9|a^2=gi,b^2=fgh,c^2=fgh,d^2=fh,e^2=fg,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, 5908153);