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 | |
1606 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)(5,6,10,9)(7,11,8,12)(13,14)(15,21)(16,24)(17,18)(19,22)(20,25)(23,26)(27,28)(29,30,34,37)(31,32,38,35)(33,41,42,36)(39,43,44,40);(5,7,10,8)(6,11,9,12)(14,18)(15,22)(20,24)(23,28)(31,38)(32,35)(39,44)(40,43);(5,8,10,7)(6,11,9,12)(13,15)(14,19)(16,23)(17,22)(18,21)(20,26)(24,27)(25,28)(29,31)(30,35)(32,37)(33,39)(34,38)(36,43)(40,41)(42,44);(1,2)(3,4)(5,9,10,6)(7,11,8,12)(13,14)(15,19)(16,24)(17,18)(20,25)(21,22)(23,27)(26,28)(29,32,34,35)(30,31,37,38)(33,40,42,43)(36,39,41,44);(13,16,17,25)(14,20,18,24)(15,23,22,28)(19,26,21,27)(29,33)(30,36)(31,39)(32,40)(34,42)(35,43)(37,41)(38,44)Polycyclic presentation:
2^9|a^2=fg,b^2=gh,c^2=gh,d^2=fg,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, 4770421);