Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| automorphism | Aut(G) order 1048576 |
| 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 | |
1806 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)(7,11)(8,10)(9,12)(13,14,17,20)(15,21,23,18)(16,19,24,26)(22,27,28,25)(30,34)(31,37)(36,40)(39,43);(5,6)(7,11)(8,10)(9,12)(16,24)(19,26)(22,28)(25,27)(29,30,33,34)(31,38,37,35)(32,40,41,36)(39,42,43,44);(5,6,8,10)(7,9,12,11)(13,15,17,23)(14,18,20,21)(16,22,24,28)(19,25,26,27)(29,30)(31,35)(32,40)(33,34)(36,41)(37,38)(39,44)(42,43);(1,2,3,4)(5,6)(7,9)(8,10)(11,12)(13,14)(15,21)(16,19)(17,20)(18,23)(22,27)(24,26)(25,28)(29,31,33,37)(30,35,34,38)(32,39,41,43)(36,42,40,44);(5,7,8,12)(6,9,10,11)(13,16)(14,19)(15,22)(17,24)(18,25)(20,26)(21,27)(23,28)(29,32,33,41)(30,36,34,40)(31,39,37,43)(35,42,38,44)Polycyclic presentation:
2^9|a^2=gh,b^2=f,c^2=fg,d^2=ghi,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5191937);