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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1443 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
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)(19,26,24,21)(20,23,30,32)(27,31,28,29)(34,35)(37,45)(39,41)(43,48);(4,8)(6,7)(11,15)(13,16)(19,24)(21,26)(27,28)(29,31)(33,34,38,35)(36,41,44,39)(37,42,45,40)(43,46,48,47);(2,3)(6,7)(9,10,12,14)(11,13,15,16)(17,19,22,24)(18,21,25,26)(20,28,30,27)(23,31,32,29)(33,35,38,34)(36,41,44,39)(37,40,45,42)(43,46,48,47);(1,3,5,2)(4,7,8,6)(9,10)(11,13)(12,14)(15,16)(17,20)(18,23)(19,27)(21,29)(22,30)(24,28)(25,32)(26,31)(33,36,38,44)(34,39,35,41)(37,43,45,48)(40,46,42,47);(1,4,5,8)(2,6,3,7)(9,11,12,15)(10,13,14,16)(17,21)(18,24)(19,25)(20,29)(22,26)(23,28)(27,32)(30,31)(33,37,38,45)(34,40,35,42)(36,43,44,48)(39,46,41,47)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=gi,d^2=fi,e^2=gh,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, 5235173);