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 | |
1724 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(13,14)(15,16)(17,25)(18,19)(20,22)(21,28)(23,27)(24,26)(29,30,34,32)(31,36,39,35)(33,37,42,40)(38,44,41,43);(6,9)(10,12)(14,19)(15,22)(21,25)(23,26)(29,31,34,39)(30,35,32,36)(33,38,42,41)(37,43,40,44);(5,6,8,9)(7,10,11,12)(13,15)(14,20)(16,19)(17,26)(18,22)(21,24)(23,28)(25,27)(29,31)(30,36)(32,35)(33,41)(34,39)(37,43)(38,42)(40,44);(1,2)(3,4)(13,16,18,20)(14,15,19,22)(17,27,28,24)(21,26,25,23)(29,32,34,30)(31,35,39,36)(33,37,42,40)(38,44,41,43);(5,7,8,11)(6,10,9,12)(13,17)(14,21)(15,23)(16,24)(18,28)(19,25)(20,27)(22,26)(29,33,34,42)(30,37,32,40)(31,38,39,41)(35,43,36,44)Polycyclic presentation:
2^9|a^2=fg,b^2=fg,c^2=fh,d^2=f,e^2=gh,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, 3261016);