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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1685 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(6,9)(10,12)(13,14)(15,19)(16,18)(17,20)(21,23)(22,27)(24,26)(25,28)(29,30,34,37)(31,38,36,33)(32,35,41,43)(39,40,44,42);(2,4)(14,16)(19,21)(20,24)(27,28)(29,31,34,36)(30,33,37,38)(32,39,41,44)(35,42,43,40);(1,2)(3,4)(13,15,18,23)(14,19,16,21)(17,25,26,22)(20,28,24,27)(29,31)(30,33)(32,39)(34,36)(35,42)(37,38)(40,43)(41,44);(5,6,8,9)(7,10,11,12)(13,16,18,14)(15,21,23,19)(17,20,26,24)(22,27,25,28)(29,32)(30,35)(31,39)(33,42)(34,41)(36,44)(37,43)(38,40);(5,7,8,11)(6,10,9,12)(13,17,18,26)(14,20,16,24)(15,22,23,25)(19,27,21,28)(29,33)(30,36)(31,37)(32,40)(34,38)(35,39)(41,42)(43,44)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=fg,d^2=i,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, 3634381);