Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4861280 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C4 |
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 | |
1438 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,5)(6,8)(9,10,12,14)(11,15,16,13)(17,18)(19,25)(20,28)(21,22)(23,27)(24,30)(26,31)(29,32)(34,35)(36,43)(40,42)(44,45);(3,7)(6,8)(11,16)(13,15)(18,22)(23,25)(24,28)(31,32)(33,34,38,35)(36,41,43,39)(37,40,46,42)(44,48,45,47);(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,23)(20,24)(21,22)(25,27)(26,31)(28,30)(29,32)(33,35,38,34)(36,41,43,39)(37,42,46,40)(44,48,45,47);(1,3,4,7)(2,6,5,8)(9,11)(10,13)(12,16)(14,15)(17,19,21,27)(18,23,22,25)(20,29,30,26)(24,32,28,31)(33,36)(34,39)(35,41)(37,45)(38,43)(40,48)(42,47)(44,46);(17,20,21,30)(18,24,22,28)(19,26,27,29)(23,31,25,32)(33,37,38,46)(34,40,35,42)(36,44,43,45)(39,47,41,48)Polycyclic presentation:
2^9|a^2=fi,b^2=fh,c^2=fh,d^2=gi,e^2=f,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, 4802698);