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 | |
1382 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(3,7)(6,8)(9,10,12,14)(11,13,15,16)(17,18)(19,25)(20,24)(21,22)(23,27)(26,32)(28,31)(29,30)(33,34)(35,40)(36,37)(38,41)(39,42)(43,47)(44,45)(46,48);(2,5)(6,8)(11,15)(13,16)(18,22)(23,25)(24,30)(31,32)(33,35,39,36)(34,40,42,37)(38,44,46,43)(41,45,48,47);(1,2,4,5)(3,6,7,8)(9,11)(10,13)(12,15)(14,16)(17,18)(19,25)(20,24)(21,22)(23,27)(26,32)(28,31)(29,30)(33,36,39,35)(34,40,42,37)(38,44,46,43)(41,47,48,45);(9,10)(11,13)(12,14)(15,16)(17,19,21,27)(18,23,22,25)(20,28,29,26)(24,32,30,31)(33,37,39,40)(34,36,42,35)(38,45,46,47)(41,44,48,43);(1,3)(2,6)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,20,21,29)(18,24,22,30)(19,26,27,28)(23,31,25,32)(33,38,39,46)(34,41,42,48)(35,43,36,44)(37,45,40,47)Polycyclic presentation:
2^9|a^2=fi,b^2=g,c^2=ghi,d^2=fg,e^2=fg,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, 5627260);