Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4861280 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
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 | |
1778 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(7,11)(9,12)(13,14,17,18)(15,21,22,19)(16,24,26,20)(23,27,25,28)(29,30)(31,36)(32,37)(33,41)(34,35)(38,43)(39,40)(42,44);(5,6,8,10)(7,9,11,12)(14,18)(15,22)(20,24)(23,25)(30,35)(31,37)(38,41)(39,44);(5,6)(7,9)(8,10)(11,12)(13,14)(15,19)(16,20)(17,18)(21,22)(23,27)(24,26)(25,28)(29,31,34,37)(30,36,35,32)(33,39,43,44)(38,42,41,40);(1,2)(3,4)(13,15,17,22)(14,19,18,21)(16,25,26,23)(20,28,24,27)(29,32,34,36)(30,37,35,31)(33,42,43,40)(38,39,41,44);(5,7)(6,9)(8,11)(10,12)(13,16,17,26)(14,20,18,24)(15,23,22,25)(19,27,21,28)(29,33,34,43)(30,38,35,41)(31,39,37,44)(32,40,36,42)Polycyclic presentation:
2^9|a^2=g,b^2=fh,c^2=fg,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4769178);