Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #6131635 |
| 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 | |
1717 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)(6,9)(7,11)(15,21)(16,23)(18,24)(19,26)(29,30,34,32)(31,35,37,38)(33,40,41,36)(39,44,43,42);(1,2,3,4)(7,11)(10,12)(15,21)(18,24)(22,27)(25,28)(31,37)(35,38)(39,43)(42,44);(5,6)(7,10)(8,9)(11,12)(13,14,17,20)(15,18,21,24)(16,19,23,26)(22,25,27,28)(29,31,34,37)(30,35,32,38)(33,39,41,43)(36,42,40,44);(5,6)(7,10)(8,9)(11,12)(13,15)(14,18)(16,22)(17,21)(19,25)(20,24)(23,27)(26,28)(29,32,34,30)(31,38,37,35)(33,40,41,36)(39,44,43,42);(5,7,8,11)(6,10,9,12)(13,16)(14,19)(15,22)(17,23)(18,25)(20,26)(21,27)(24,28)(29,33)(30,36)(31,39)(32,40)(34,41)(35,42)(37,43)(38,44)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=ghi,d^2=h,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dgi,e^b=eg68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6039786);