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 | |
1300 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(3,8)(4,6)(9,10,13,11)(12,15,16,14)(20,27)(24,29)(25,31)(30,32)(33,34,38,39)(35,36,41,40)(37,42,45,47)(43,44,48,46);(1,2,5,7)(3,4,8,6)(9,10,13,11)(12,14,16,15)(18,22)(23,28)(24,29)(30,32)(34,39)(35,41)(37,45)(44,46);(1,3,5,8)(2,6,7,4)(9,11,13,10)(12,15,16,14)(17,18)(19,23)(20,24)(21,22)(25,30)(26,28)(27,29)(31,32)(33,35,38,41)(34,40,39,36)(37,43,45,48)(42,46,47,44);(1,3)(2,4)(5,8)(6,7)(9,12,13,16)(10,14,11,15)(17,19,21,26)(18,23,22,28)(20,25,27,31)(24,30,29,32)(33,36,38,40)(34,41,39,35)(37,46,45,44)(42,43,47,48);(1,4)(2,3)(5,6)(7,8)(17,20,21,27)(18,24,22,29)(19,25,26,31)(23,30,28,32)(33,37)(34,42)(35,43)(36,44)(38,45)(39,47)(40,46)(41,48)Polycyclic presentation:
2^9|a^2=gi,b^2=i,c^2=fi,d^2=hi,e^2=gh,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, 5854855);