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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1583 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)(9,10,12,14)(11,13,15,16)(17,18)(19,22)(20,25)(21,24)(23,28)(26,29)(27,31)(30,32)(33,34,38,36)(35,40,43,39)(37,44,46,41)(42,47,45,48);(2,5)(6,8)(11,15)(13,16)(18,19)(21,29)(23,25)(27,32)(33,35,38,43)(34,39,36,40)(37,45,46,42)(41,48,44,47);(9,11)(10,13)(12,15)(14,16)(17,19,22,18)(20,25,28,23)(21,26,29,24)(27,31,32,30)(33,35)(34,40)(36,39)(37,42)(38,43)(41,48)(44,47)(45,46);(1,2)(3,6)(4,5)(7,8)(17,20)(18,23)(19,25)(21,27)(22,28)(24,30)(26,31)(29,32)(33,36,38,34)(35,40,43,39)(37,44,46,41)(42,48,45,47);(1,3)(2,6)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,21)(18,24)(19,26)(20,27)(22,29)(23,30)(25,31)(28,32)(33,37,38,46)(34,41,36,44)(35,42,43,45)(39,47,40,48)Polycyclic presentation:
2^9|a^2=gh,b^2=fg,c^2=g,d^2=fg,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, 5865693);