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 | |
1296 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)(10,12)(11,16)(17,18,21,24)(19,25,27,22)(20,28,29,23)(26,30,32,31)(33,34)(35,36)(37,41)(38,39)(40,43)(42,44)(45,46)(47,48);(2,5)(6,8)(9,10,14,12)(11,13,16,15)(34,39)(36,40)(37,45)(42,48);(9,11,14,16)(10,15,12,13)(17,19,21,27)(18,22,24,25)(20,26,29,32)(23,30,28,31)(33,35)(34,40)(36,39)(37,42)(38,43)(41,47)(44,46)(45,48);(1,2)(3,6)(4,5)(7,8)(9,12,14,10)(11,13,16,15)(17,18)(19,25)(20,23)(21,24)(22,27)(26,31)(28,29)(30,32)(33,36,38,40)(34,35,39,43)(37,44,45,47)(41,42,46,48);(1,3)(2,6)(4,7)(5,8)(9,13)(10,11)(12,16)(14,15)(17,20,21,29)(18,23,24,28)(19,26,27,32)(22,30,25,31)(33,37)(34,41)(35,42)(36,44)(38,45)(39,46)(40,47)(43,48)Polycyclic presentation:
2^9|a^2=fg,b^2=h,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, 5870336);