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 | |
1358 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,6,4)(3,7,5,8)(11,12)(15,16)(17,18,22,19)(20,23,28,25)(21,24,30,26)(27,31,29,32)(33,34,37,40)(35,41,42,38)(36,44,45,39)(43,46,48,47);(1,3,6,5)(2,7,4,8)(10,14)(15,16)(18,19)(23,25)(24,26)(31,32)(35,42)(36,45)(38,41)(39,44);(1,4,6,2)(3,7,5,8)(9,10)(11,15)(12,16)(13,14)(17,19,22,18)(20,23,28,25)(21,26,30,24)(27,31,29,32)(33,35,37,42)(34,38,40,41)(36,43,45,48)(39,46,44,47);(9,11,13,12)(10,15,14,16)(17,20,22,28)(18,23,19,25)(21,29,30,27)(24,32,26,31)(33,34)(35,41)(36,39)(37,40)(38,42)(43,47)(44,45)(46,48);(1,5,6,3)(2,7,4,8)(9,12,13,11)(10,16,14,15)(17,21,22,30)(18,24,19,26)(20,27,28,29)(23,31,25,32)(33,36,37,45)(34,39,40,44)(35,43,42,48)(38,46,41,47)Polycyclic presentation:
2^9|a^2=ghi,b^2=fhi,c^2=ghi,d^2=fgi,e^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, 5853709);