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 | |
1433 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)(3,6)(4,5)(7,8)(10,14)(12,15)(18,22)(19,25)(20,27)(30,32)(35,41)(36,43)(39,44)(40,45);(1,2,5,4)(3,7,8,6)(9,10)(11,15)(12,16)(13,14)(18,22)(19,25)(24,29)(26,31)(34,38)(39,44)(40,45)(46,48);(1,3)(2,6)(4,7)(5,8)(9,11)(10,12)(13,16)(14,15)(17,18)(19,23)(20,24)(21,22)(25,28)(26,30)(27,29)(31,32)(33,34,37,38)(35,39,41,44)(36,40,43,45)(42,46,47,48);(1,3)(2,7)(4,6)(5,8)(9,11)(10,15)(12,14)(13,16)(17,19,21,25)(18,23,22,28)(20,26,27,31)(24,30,29,32)(33,35)(34,39)(36,42)(37,41)(38,44)(40,46)(43,47)(45,48);(1,4,5,2)(3,7,8,6)(9,12,13,15)(10,16,14,11)(17,20)(18,24)(19,26)(21,27)(22,29)(23,30)(25,31)(28,32)(33,36,37,43)(34,40,38,45)(35,42,41,47)(39,46,44,48)Polycyclic presentation:
2^9|b^2=fhi,c^2=fgh,d^2=g,e^2=gh,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, 4783978);