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 | |
1298 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,6)(3,7,8,5)(10,13)(14,15)(18,22)(19,25)(20,28)(30,32)(33,34,37,38)(35,39,43,41)(36,44,45,40)(42,47,48,46);(3,8)(5,7)(9,10,12,13)(11,15,16,14)(34,38)(36,45)(39,41)(42,48);(9,10)(11,14)(12,13)(15,16)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31)(33,35,37,43)(34,39,38,41)(36,42,45,48)(40,46,44,47);(1,3)(2,5)(4,8)(6,7)(9,11,12,16)(10,14,13,15)(17,19)(18,23)(20,27)(21,25)(22,26)(24,30)(28,31)(29,32)(33,34)(35,41)(36,40)(37,38)(39,43)(42,47)(44,45)(46,48);(1,2)(3,5)(4,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,20,21,28)(18,24,22,29)(19,27,25,31)(23,30,26,32)(33,36,37,45)(34,40,38,44)(35,42,43,48)(39,46,41,47)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,c^2=fh,d^2=h,e^2=fg,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, 5908871);