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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1371 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,7)(3,6,8,4)(9,10,13,11)(12,15,16,14)(18,22)(23,28)(24,29)(30,32)(33,34)(35,41)(36,43)(37,40)(38,39)(42,45)(44,48)(46,47);(1,2)(3,4)(5,7)(6,8)(9,10,13,11)(12,14,16,15)(19,25)(23,28)(26,31)(30,32)(34,39)(36,41)(40,47)(44,45);(1,3)(2,6)(4,7)(5,8)(9,11,13,10)(12,15,16,14)(17,18,21,22)(19,23,25,28)(20,24,27,29)(26,30,31,32)(33,35)(34,36)(37,42)(38,43)(39,41)(40,44)(45,47)(46,48);(1,4,5,6)(2,3,7,8)(9,12)(10,14)(11,15)(13,16)(17,19)(18,23)(20,26)(21,25)(22,28)(24,30)(27,31)(29,32)(33,36)(34,35)(37,45)(38,41)(39,43)(40,48)(42,47)(44,46);(1,3)(2,4)(5,8)(6,7)(17,20,21,27)(18,24,22,29)(19,26,25,31)(23,30,28,32)(33,37,38,46)(34,40,39,47)(35,42,43,48)(36,44,41,45)Polycyclic presentation:
2^9|a^2=fi,b^2=fhi,c^2=gi,d^2=h,e^2=g,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, 4796455);