Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4417862 |
| 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 | |
1754 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,3,4)(5,6,9,8)(7,11,12,10)(15,21)(16,23)(19,24)(20,25)(29,30,33,34)(31,35,38,42)(32,39,41,36)(37,44,40,43);(5,6,9,8)(7,10,12,11)(14,18)(19,24)(20,25)(26,28)(30,34)(35,42)(36,39)(43,44);(5,7,9,12)(6,10,8,11)(13,14)(15,19)(16,20)(17,18)(21,24)(22,26)(23,25)(27,28)(29,30)(31,35)(32,36)(33,34)(37,43)(38,42)(39,41)(40,44);(1,2)(3,4)(5,8,9,6)(7,11,12,10)(13,15)(14,19)(16,22)(17,21)(18,24)(20,26)(23,27)(25,28)(29,31,33,38)(30,35,34,42)(32,40,41,37)(36,44,39,43);(13,16,17,23)(14,20,18,25)(15,22,21,27)(19,26,24,28)(29,32)(30,36)(31,37)(33,41)(34,39)(35,43)(38,40)(42,44)Polycyclic presentation:
2^9|a^2=fghi,b^2=fg,c^2=fg,d^2=g,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4330136);