Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5473759 |
| 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 | |
1350 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,6)(3,5,7,8)(9,10,14,12)(11,13,16,15)(20,27)(24,29)(25,31)(30,32)(33,34,38,35)(36,39,44,41)(37,40,45,42)(43,46,48,47);(9,11,14,16)(10,15,12,13)(18,22)(23,28)(24,29)(30,32)(34,35)(37,45)(39,41)(43,48);(10,12)(13,15)(17,18)(19,23)(20,24)(21,22)(25,30)(26,28)(27,29)(31,32)(33,35,38,34)(36,41,44,39)(37,40,45,42)(43,46,48,47);(1,2)(3,5)(4,6)(7,8)(9,12,14,10)(11,13,16,15)(17,19,21,26)(18,23,22,28)(20,25,27,31)(24,30,29,32)(33,36,38,44)(34,39,35,41)(37,43,45,48)(40,46,42,47);(1,3,4,7)(2,5,6,8)(9,13)(10,11)(12,16)(14,15)(17,20)(18,24)(19,25)(21,27)(22,29)(23,30)(26,31)(28,32)(33,37,38,45)(34,40,35,42)(36,43,44,48)(39,46,41,47)Polycyclic presentation:
2^9|a^2=fi,b^2=g,c^2=fg,d^2=gh,e^2=fgi,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5489336);