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) = C26×C4 |
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 | |
1834 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)(6,9)(10,12)(15,21)(16,23)(18,24)(19,26)(30,32)(31,37)(33,41)(42,44);(13,14,17,20)(15,18,21,24)(16,19,23,26)(22,25,27,28)(29,30,34,32)(31,38,37,35)(33,36,41,40)(39,44,43,42);(5,6)(7,10)(8,9)(11,12)(13,14)(15,18)(16,19)(17,20)(21,24)(22,25)(23,26)(27,28)(29,31,34,37)(30,35,32,38)(33,39,41,43)(36,42,40,44);(1,2)(3,4)(5,7,8,11)(6,10,9,12)(13,15)(14,18)(16,22)(17,21)(19,25)(20,24)(23,27)(26,28)(29,32,34,30)(31,38,37,35)(33,36,41,40)(39,42,43,44);(13,16,17,23)(14,19,20,26)(15,22,21,27)(18,25,24,28)(29,33)(30,36)(31,39)(32,40)(34,41)(35,42)(37,43)(38,44)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=f,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, 4330267);