Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| 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 | |
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:
(5,6,9,8)(7,11,12,10)(14,18)(15,21)(16,23)(26,28)(29,30)(31,37)(32,35)(33,36)(34,39)(38,44)(40,42)(41,43);(2,4)(14,18)(15,21)(20,25)(22,27)(29,30,33,36)(31,37,39,34)(32,35,42,40)(38,44,41,43);(1,2,3,4)(5,6,9,8)(7,10,12,11)(13,14)(15,19)(16,20)(17,18)(21,24)(22,26)(23,25)(27,28)(29,30)(31,34)(32,40)(33,36)(35,42)(37,39)(38,44)(41,43);(5,7,9,12)(6,10,8,11)(13,15,17,21)(14,19,18,24)(16,22,23,27)(20,26,25,28)(29,31)(30,34)(32,41)(33,39)(35,44)(36,37)(38,42)(40,43);(5,8,9,6)(7,11,12,10)(13,16)(14,20)(15,22)(17,23)(18,25)(19,26)(21,27)(24,28)(29,32)(30,35)(31,38)(33,42)(34,43)(36,40)(37,44)(39,41)Polycyclic presentation:
2^9|a^2=fgi,b^2=f,c^2=hi,d^2=fi,e^2=fgi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3737124);