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 | |
1708 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)(13,14,18,21)(15,16,22,19)(17,25,26,20)(23,27,28,24)(29,30)(31,33)(32,37)(34,35)(36,38)(39,42)(40,44)(41,43);(1,2,3,4)(15,22)(16,19)(23,28)(24,27)(30,35)(31,38)(37,43)(39,44);(1,2)(3,4)(13,15)(14,19)(16,21)(17,23)(18,22)(20,27)(24,25)(26,28)(29,31)(30,36)(32,39)(33,35)(34,38)(37,40)(41,44)(42,43);(5,6,8,9)(7,10,11,12)(13,16,18,19)(14,15,21,22)(17,24,26,27)(20,23,25,28)(29,32)(30,37)(31,39)(33,42)(34,41)(35,43)(36,40)(38,44);(5,7,8,11)(6,10,9,12)(13,17)(14,20)(15,23)(16,24)(18,26)(19,27)(21,25)(22,28)(29,33,34,36)(30,38,35,31)(32,40,41,42)(37,39,43,44)Polycyclic presentation:
2^9|a^2=fg,b^2=gh,d^2=i,e^2=gi,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, 3636810);