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 | |
1320 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)(3,8)(4,7)(5,6)(11,15)(13,16)(18,22)(19,25)(24,29)(27,31)(33,34,38,42)(35,43,44,39)(36,46,47,40)(37,45,48,41);(2,6)(3,7)(9,10,12,14)(11,13,15,16)(20,28)(24,29)(27,31)(30,32)(35,44)(37,48)(39,43)(41,45);(1,3)(2,4)(5,7)(6,8)(17,18,21,22)(19,26,25,23)(20,24,28,29)(27,32,31,30)(33,35,38,44)(34,39,42,43)(36,41,47,45)(37,46,48,40);(1,4)(2,7)(3,6)(5,8)(9,11)(10,13)(12,15)(14,16)(17,19,21,25)(18,23,22,26)(20,27,28,31)(24,30,29,32)(33,36,38,47)(34,40,42,46)(35,45,44,41)(37,39,48,43);(1,4)(2,7)(3,6)(5,8)(17,20)(18,24)(19,27)(21,28)(22,29)(23,30)(25,31)(26,32)(33,37,38,48)(34,41,42,45)(35,46,44,40)(36,39,47,43)Polycyclic presentation:
2^9|a^2=fh,b^2=fgi,c^2=fg,d^2=fg,e^2=fh,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, 5392926);