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 | |
1316 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)(3,4)(5,6)(7,8)(9,10,13,14)(11,12,15,16)(19,25)(20,27)(22,28)(23,30)(33,34,37,38)(35,41,43,39)(36,40,44,45)(42,47,48,46);(1,3)(2,6)(4,8)(5,7)(9,11,13,15)(10,12,14,16)(19,25)(22,28)(26,31)(29,32)(34,38)(36,44)(39,41)(42,48);(1,4,7,6)(2,3,8,5)(9,11)(10,12)(13,15)(14,16)(17,18,21,24)(19,22,25,28)(20,23,27,30)(26,29,31,32)(33,34)(35,39)(36,40)(37,38)(41,43)(42,46)(44,45)(47,48);(1,5,7,3)(2,4,8,6)(9,12)(10,11)(13,16)(14,15)(17,19)(18,22)(20,26)(21,25)(23,29)(24,28)(27,31)(30,32)(33,35)(34,39)(36,42)(37,43)(38,41)(40,46)(44,48)(45,47);(1,6,7,4)(2,3,8,5)(17,20)(18,23)(19,26)(21,27)(22,29)(24,30)(25,31)(28,32)(33,36,37,44)(34,40,38,45)(35,42,43,48)(39,46,41,47)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=hi,d^2=fg,e^2=f,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, 5926668);