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 | |
1648 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6,8,10)(7,9,11,12)(13,14)(15,18)(16,21)(17,22)(19,24)(20,26)(23,27)(25,28)(29,30)(31,37)(32,35)(33,36)(34,39)(38,44)(40,42)(41,43);(2,4)(14,15)(19,21)(20,22)(27,28)(29,30,33,36)(31,37,39,34)(32,35,42,40)(38,44,41,43);(1,2)(3,4)(13,15,18,14)(16,21,24,19)(17,20,26,22)(23,27,25,28)(29,30)(31,34)(32,40)(33,36)(35,42)(37,39)(38,44)(41,43);(5,6)(7,9)(8,10)(11,12)(13,16)(14,19)(15,21)(17,25)(18,24)(20,28)(22,27)(23,26)(29,31,33,39)(30,34,36,37)(32,41,42,38)(35,44,40,43);(5,7,8,11)(6,9,10,12)(13,17,18,26)(14,20,15,22)(16,23,24,25)(19,27,21,28)(29,32,33,42)(30,35,36,40)(31,38,39,41)(34,43,37,44)Polycyclic presentation:
2^9|a^2=fi,b^2=fg,c^2=g,d^2=fg,e^2=i,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, 3536308);