Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4131229 |
| 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 | |
1474 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(3,7)(6,8)(9,10,13,12)(11,14,16,15)(18,22)(19,25)(20,28)(30,32)(33,34)(35,41)(36,40)(37,38)(39,43)(42,48)(44,46)(45,47);(1,2,5,4)(3,6,7,8)(9,11,13,16)(10,14,12,15)(18,22)(20,28)(23,26)(27,31)(34,38)(39,41)(40,44)(47,48);(9,11)(10,14)(12,15)(13,16)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31)(33,34)(35,39)(36,44)(37,38)(40,46)(41,43)(42,48)(45,47);(1,3)(2,6)(4,8)(5,7)(17,19)(18,23)(20,27)(21,25)(22,26)(24,30)(28,31)(29,32)(33,35,37,43)(34,39,38,41)(36,45,46,42)(40,48,44,47);(1,4,5,2)(3,8,7,6)(9,12,13,10)(11,15,16,14)(17,20,21,28)(18,24,22,29)(19,27,25,31)(23,30,26,32)(33,36,37,46)(34,40,38,44)(35,42,43,45)(39,47,41,48)Polycyclic presentation:
2^9|a^2=fgh,b^2=hi,d^2=f,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4191466);