Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| 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 | |
1594 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)(5,6)(7,12)(8,11)(9,10)(13,14,17,18)(15,21,22,19)(16,20,25,24)(23,27,28,26)(29,30)(31,36)(32,37)(33,38)(34,35)(39,44)(40,41)(42,43);(5,7,10,8)(6,11,9,12)(14,18)(15,22)(20,24)(23,28)(30,35)(31,37)(38,43)(39,41);(5,8,10,7)(6,11,9,12)(13,14)(15,19)(16,24)(17,18)(20,25)(21,22)(23,27)(26,28)(29,31,34,37)(30,36,35,32)(33,41,42,39)(38,40,43,44);(1,2)(3,4)(13,15,17,22)(14,19,18,21)(16,23,25,28)(20,26,24,27)(29,32,34,36)(30,37,35,31)(33,40,42,44)(38,41,43,39);(5,9,10,6)(7,12,8,11)(13,16,17,25)(14,20,18,24)(15,23,22,28)(19,26,21,27)(29,33,34,42)(30,38,35,43)(31,39,37,41)(32,40,36,44)Polycyclic presentation:
2^9|a^2=g,b^2=fh,c^2=fg,d^2=h,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4507858);