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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1300 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,5)(3,7)(9,10)(11,12)(13,14)(15,16)(17,18,21,24)(19,25,26,22)(20,23,28,30)(27,31,32,29)(33,34,38,41)(35,39,36,43)(37,45,46,40)(42,48,44,47);(2,5)(6,8)(10,14)(11,16)(19,26)(22,25)(27,32)(29,31)(33,35,38,36)(34,39,41,43)(37,42,46,44)(40,47,45,48);(9,11,13,16)(10,15,14,12)(17,19)(18,22)(20,27)(21,26)(23,29)(24,25)(28,32)(30,31)(33,36,38,35)(34,39,41,43)(37,44,46,42)(40,47,45,48);(1,2)(3,6)(4,5)(7,8)(9,12,13,15)(10,16,14,11)(17,18)(19,25)(20,23)(21,24)(22,26)(27,31)(28,30)(29,32)(33,34)(35,39)(36,43)(37,40)(38,41)(42,47)(44,48)(45,46);(1,3)(2,6)(4,7)(5,8)(9,10)(11,15)(12,16)(13,14)(17,20,21,28)(18,23,24,30)(19,27,26,32)(22,29,25,31)(33,37,38,46)(34,40,41,45)(35,42,36,44)(39,47,43,48)Polycyclic presentation:
2^9|a^2=fg,b^2=gh,c^2=fgh,d^2=f,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, 5870400);