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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1769 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6,9,12)(7,10,8,11)(14,18)(15,21)(16,23)(26,28)(29,30)(31,33)(32,39)(34,35)(36,41)(37,43)(38,40)(42,44);(5,7,9,8)(6,10,12,11)(14,18)(19,24)(20,25)(26,28)(30,35)(31,37)(36,39)(38,44);(5,6)(7,10)(8,11)(9,12)(13,14)(15,19)(16,20)(17,18)(21,24)(22,26)(23,25)(27,28)(29,31,34,37)(30,33,35,43)(32,38,41,44)(36,42,39,40);(1,2,3,4)(5,8,9,7)(6,11,12,10)(13,15)(14,19)(16,22)(17,21)(18,24)(20,26)(23,27)(25,28)(29,32,34,41)(30,36,35,39)(31,38,37,44)(33,42,43,40);(13,16,17,23)(14,20,18,25)(15,22,21,27)(19,26,24,28)(29,33)(30,37)(31,35)(32,40)(34,43)(36,38)(39,44)(41,42)Polycyclic presentation:
2^9|a^2=fgh,b^2=fgh,c^2=f,d^2=gi,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, 4342531);