Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4417862 |
| 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 | |
1368 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,6)(3,8)(11,15)(14,16)(17,18)(19,22)(20,23)(21,26)(24,30)(25,28)(27,32)(29,31)(33,34,37,40)(35,41,42,38)(36,44,45,39)(43,46,48,47);(9,10,13,12)(11,14,15,16)(18,19)(23,25)(24,26)(31,32)(35,42)(38,41)(43,48)(46,47);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,16)(13,15)(17,19,22,18)(20,25,28,23)(21,26,30,24)(27,32,29,31)(33,35)(34,38)(36,43)(37,42)(39,46)(40,41)(44,47)(45,48);(1,3,5,8)(2,7,6,4)(9,10,13,12)(11,14,15,16)(17,20,22,28)(18,23,19,25)(21,29,30,27)(24,32,26,31)(33,36,37,45)(34,39,40,44)(35,43,42,48)(38,46,41,47);(1,4,5,7)(2,8,6,3)(9,12,13,10)(11,16,15,14)(17,21)(18,24)(19,26)(20,27)(22,30)(23,31)(25,32)(28,29)(33,34)(35,38)(36,39)(37,40)(41,42)(43,46)(44,45)(47,48)Polycyclic presentation:
2^9|a^2=fh,b^2=fgi,c^2=f,d^2=g,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4334848);