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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1336 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)(9,10,13,11)(12,15,16,14)(17,18,22,20)(19,23,25,26)(21,28,29,24)(27,32,31,30)(33,34)(35,40)(36,41)(37,42)(38,39)(43,45)(44,47)(46,48);(3,7)(6,8)(9,10,13,11)(12,14,16,15)(19,25)(23,26)(27,31)(30,32)(34,39)(35,42)(41,48)(43,44);(1,2,4,5)(3,6,7,8)(9,11,13,10)(12,15,16,14)(17,19,22,25)(18,23,20,26)(21,27,29,31)(24,30,28,32)(33,35,38,42)(34,40,39,37)(36,44,46,43)(41,47,48,45);(1,2)(3,6)(4,5)(7,8)(9,12)(10,14)(11,15)(13,16)(17,20,22,18)(19,26,25,23)(21,28,29,24)(27,32,31,30)(33,36,38,46)(34,41,39,48)(35,43,42,44)(37,47,40,45);(1,3)(2,6)(4,7)(5,8)(17,21)(18,24)(19,27)(20,28)(22,29)(23,30)(25,31)(26,32)(33,37,38,40)(34,42,39,35)(36,45,46,47)(41,43,48,44)Polycyclic presentation:
2^9|a^2=fghi,b^2=gi,c^2=hi,d^2=h,e^2=f,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, 5701140);