Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4861280 |
| 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 | |
1352 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,8)(3,4,7,6)(9,10,14,16)(11,12,15,13)(18,22)(19,25)(20,27)(30,32)(33,34,38,37)(35,41,43,39)(36,40,44,45)(42,47,48,46);(1,3,5,7)(2,6,8,4)(9,11)(10,13)(12,16)(14,15)(18,22)(19,25)(24,29)(26,31)(36,44)(40,45)(42,48)(46,47);(1,4)(2,7)(3,8)(5,6)(9,12,14,13)(10,15,16,11)(17,18,21,22)(19,23,25,28)(20,24,27,29)(26,30,31,32)(33,35,38,43)(34,39,37,41)(36,42,44,48)(40,46,45,47);(1,4)(2,3)(5,6)(7,8)(9,11)(10,12)(13,16)(14,15)(17,19)(18,23)(20,26)(21,25)(22,28)(24,30)(27,31)(29,32)(33,36,38,44)(34,40,37,45)(35,42,43,48)(39,46,41,47);(1,2)(3,6)(4,7)(5,8)(9,13,14,12)(10,15,16,11)(17,20)(18,24)(19,26)(21,27)(22,29)(23,30)(25,31)(28,32)(33,37,38,34)(35,41,43,39)(36,45,44,40)(42,47,48,46)Polycyclic presentation:
2^9|a^2=gh,b^2=fhi,c^2=ghi,d^2=fgh,e^2=fgi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4835910);