Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| 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 | |
1402 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)(3,7)(4,5)(6,8)(9,10,13,11)(12,15,16,14)(18,22)(19,25)(24,29)(27,31)(33,34,38,36)(35,39,41,42)(37,44,45,40)(43,48,47,46);(1,3,5,8)(2,6,4,7)(9,10,13,11)(12,14,16,15)(20,28)(24,29)(27,31)(30,32)(35,41)(39,42)(43,47)(46,48);(1,2,5,4)(3,7,8,6)(9,11,13,10)(12,15,16,14)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31)(33,35,38,41)(34,39,36,42)(37,43,45,47)(40,46,44,48);(1,4,5,2)(3,7,8,6)(9,12)(10,14)(11,15)(13,16)(17,19)(18,23)(20,27)(21,25)(22,26)(24,30)(28,31)(29,32)(33,36,38,34)(35,42,41,39)(37,44,45,40)(43,48,47,46);(1,3)(2,7)(4,6)(5,8)(17,20)(18,24)(19,27)(21,28)(22,29)(23,30)(25,31)(26,32)(33,37)(34,40)(35,43)(36,44)(38,45)(39,46)(41,47)(42,48)Polycyclic presentation:
2^9|a^2=fghi,b^2=fgi,c^2=ghi,d^2=h,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5701254);