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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1756 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)(5,6,8,9)(7,10,12,11)(13,14)(15,20)(16,25)(17,23)(18,19)(21,26)(22,28)(24,27)(29,30,34,38)(31,36,32,35)(33,41,42,37)(39,43,40,44);(6,9)(10,11)(14,19)(16,26)(20,27)(22,23)(29,31,34,32)(30,35,38,36)(33,40,42,39)(37,44,41,43);(5,6,8,9)(7,11,12,10)(13,15,18,24)(14,20,19,27)(16,22,26,23)(17,25,28,21)(29,32,34,31)(30,36,38,35)(33,40,42,39)(37,44,41,43);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,16)(14,21)(15,23)(17,27)(18,26)(19,25)(20,28)(22,24)(29,33,34,42)(30,37,38,41)(31,39,32,40)(35,43,36,44);(5,7)(6,10)(8,12)(9,11)(13,17,18,28)(14,22,19,23)(15,21,24,25)(16,27,26,20)(29,30)(31,36)(32,35)(33,37)(34,38)(39,44)(40,43)(41,42)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=f,d^2=fgi,e^2=fgh,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, 5190359);