Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5956507 |
| 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 | |
1306 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)(4,5)(6,8)(9,10)(11,14)(12,16)(13,15)(18,22)(23,26)(24,29)(30,32)(33,34,37,38)(35,41,43,39)(36,40,44,45)(42,47,48,46);(1,3)(2,6)(4,7)(5,8)(9,11,13,16)(10,12,15,14)(19,25)(23,26)(27,31)(30,32)(34,38)(36,44)(39,41)(42,48);(1,3,5,8)(2,7,4,6)(9,11,13,16)(10,14,15,12)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31)(33,34)(35,39)(36,40)(37,38)(41,43)(42,46)(44,45)(47,48);(1,3)(2,7)(4,6)(5,8)(9,11)(10,12)(13,16)(14,15)(17,19)(18,23)(20,27)(21,25)(22,26)(24,30)(28,31)(29,32)(33,35,37,43)(34,39,38,41)(36,42,44,48)(40,46,45,47);(1,4,5,2)(3,6,8,7)(9,12)(10,11)(13,14)(15,16)(17,20,21,28)(18,24,22,29)(19,27,25,31)(23,30,26,32)(33,36)(34,40)(35,42)(37,44)(38,45)(39,46)(41,47)(43,48)Polycyclic presentation:
2^9|a^2=fhi,b^2=fh,c^2=hi,d^2=fhi,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dgi,e^b=ef,d^c=dfg62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6038586);