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 | |
1362 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)(3,7)(4,6)(5,8)(10,13)(14,15)(18,22)(19,25)(24,29)(27,31)(33,34,37,38)(35,39,43,42)(36,40,44,45)(41,46,48,47);(1,3)(2,5)(4,8)(6,7)(9,10,12,13)(11,14,16,15)(20,28)(24,29)(27,31)(30,32)(34,38)(39,42)(40,45)(46,47);(2,6)(3,8)(9,10)(11,15)(12,13)(14,16)(17,18,21,22)(19,26,25,23)(20,24,28,29)(27,32,31,30)(33,35,37,43)(34,39,38,42)(36,41,44,48)(40,46,45,47);(1,3)(2,5)(4,8)(6,7)(9,11)(10,14)(12,16)(13,15)(17,19,21,25)(18,23,22,26)(20,27,28,31)(24,30,29,32)(33,36,37,44)(34,40,38,45)(35,41,43,48)(39,46,42,47);(1,2)(3,5)(4,6)(7,8)(9,11)(10,14)(12,16)(13,15)(17,20,21,28)(18,24,22,29)(19,27,25,31)(23,30,26,32)(33,34)(35,42)(36,40)(37,38)(39,43)(41,47)(44,45)(46,48)Polycyclic presentation:
2^9|a^2=fh,b^2=gi,c^2=fg,d^2=fg,e^2=gh,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, 5393420);