Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4932388 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C4 |
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 | |
1362 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,5)(4,6)(7,8)(9,10)(11,12)(13,16)(14,15)(18,22)(19,25)(20,28)(30,32)(33,34,37,38)(35,39,42,44)(36,40,43,45)(41,46,47,48);(9,11,14,13)(10,12,15,16)(18,22)(23,26)(24,29)(30,32)(34,38)(36,43)(39,44)(41,47);(2,3)(6,7)(10,15)(12,16)(17,18,21,22)(19,26,25,23)(20,24,28,29)(27,32,31,30)(33,34)(35,39)(36,40)(37,38)(41,46)(42,44)(43,45)(47,48);(1,3,5,2)(4,7,8,6)(9,12,14,16)(10,11,15,13)(17,19)(18,23)(20,27)(21,25)(22,26)(24,30)(28,31)(29,32)(33,35,37,42)(34,39,38,44)(36,41,43,47)(40,46,45,48);(1,4,5,8)(2,6,3,7)(9,13,14,11)(10,16,15,12)(17,20,21,28)(18,24,22,29)(19,27,25,31)(23,30,26,32)(33,36,37,43)(34,40,38,45)(35,41,42,47)(39,46,44,48)Polycyclic presentation:
2^9|a^2=f,b^2=fgi,c^2=fh,d^2=fgh,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=ef,d^c=dfg80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4861315);