Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| 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 | |
1390 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)(3,6)(4,5)(7,8)(9,10,12,14)(11,13,15,16)(17,18,22,19)(20,25,27,23)(21,24,29,26)(28,31,32,30)(33,34)(35,39)(36,38)(37,44)(40,46)(41,48)(42,43)(45,47);(3,7)(6,8)(9,11,12,15)(10,13,14,16)(18,19)(20,27)(24,26)(28,32)(34,36)(39,43)(40,44)(47,48);(2,5)(6,8)(10,14)(13,16)(17,19,22,18)(20,23,27,25)(21,24,29,26)(28,31,32,30)(33,35)(34,39)(36,43)(37,45)(38,42)(40,48)(41,46)(44,47);(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,20,22,27)(18,23,19,25)(21,28,29,32)(24,30,26,31)(33,36,38,34)(35,39,42,43)(37,44,46,40)(41,47,45,48);(1,3)(2,6)(4,7)(5,8)(9,11)(10,13)(12,15)(14,16)(17,21,22,29)(18,24,19,26)(20,28,27,32)(23,30,25,31)(33,37)(34,40)(35,41)(36,44)(38,46)(39,47)(42,45)(43,48)Polycyclic presentation:
2^9|a^2=fhi,b^2=fi,c^2=h,d^2=f,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5150566);