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 | |
1356 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)(11,13)(12,14)(15,16)(17,18,22,25)(19,21,26,23)(20,28,31,24)(27,32,29,30)(34,38)(35,41)(40,44)(43,47);(4,8)(6,7)(11,15)(13,16)(19,26)(21,23)(27,29)(30,32)(33,34,37,38)(35,42,41,39)(36,44,45,40)(43,46,47,48);(2,3)(6,7)(9,10,12,14)(11,13,15,16)(17,19)(18,23)(20,29)(21,25)(22,26)(24,30)(27,31)(28,32)(33,34)(35,39)(36,44)(37,38)(40,45)(41,42)(43,48)(46,47);(1,3,5,2)(4,7,8,6)(9,10)(11,13)(12,14)(15,16)(17,20,22,31)(18,24,25,28)(19,27,26,29)(21,30,23,32)(33,35,37,41)(34,39,38,42)(36,43,45,47)(40,46,44,48);(1,4)(2,6)(3,7)(5,8)(9,11)(10,13)(12,15)(14,16)(17,21,22,23)(18,19,25,26)(20,30,31,32)(24,27,28,29)(33,36,37,45)(34,40,38,44)(35,43,41,47)(39,46,42,48)Polycyclic presentation:
2^9|a^2=fh,b^2=f,c^2=gi,d^2=i,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, 5234654);