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 | |
1416 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,6)(3,5,7,8)(12,16)(14,15)(18,22)(19,25)(20,28)(30,32)(33,34)(35,38)(36,41)(37,40)(39,44)(42,45)(43,48)(46,47);(1,3,4,7)(2,5,6,8)(9,10,13,11)(12,14,16,15)(18,22)(20,28)(23,26)(27,31)(34,35)(37,45)(39,41)(43,46);(1,3)(2,5)(4,7)(6,8)(9,11,13,10)(12,15,16,14)(17,18,21,22)(19,26,25,23)(20,24,28,29)(27,32,31,30)(33,35,38,34)(36,41,44,39)(37,42,45,40)(43,48,46,47);(1,2)(3,5)(4,6)(7,8)(9,12,13,16)(10,14,11,15)(17,19)(18,23)(20,27)(21,25)(22,26)(24,30)(28,31)(29,32)(33,36)(34,39)(35,41)(37,46)(38,44)(40,48)(42,47)(43,45);(1,2)(3,5)(4,6)(7,8)(17,20)(18,24)(19,27)(21,28)(22,29)(23,30)(25,31)(26,32)(33,37,38,45)(34,40,35,42)(36,43,44,46)(39,47,41,48)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,c^2=ghi,d^2=hi,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, 5810515);