Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4465744 |
| 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 | |
1834 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)(6,9)(7,11)(13,14,17,20)(15,21,22,18)(16,19,24,26)(23,27,28,25)(29,30,35,38)(31,36,32,33)(34,37,42,44)(39,43,40,41);(5,6)(7,12)(8,9)(10,11)(15,22)(18,21)(23,28)(25,27)(29,31,35,32)(30,33,38,36)(34,40,42,39)(37,43,44,41);(6,9)(10,12)(13,14)(15,18)(16,19)(17,20)(21,22)(23,25)(24,26)(27,28)(29,32,35,31)(30,36,38,33)(34,40,42,39)(37,43,44,41);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,18)(16,23)(17,22)(19,25)(20,21)(24,28)(26,27)(29,33,35,36)(30,32,38,31)(34,41,42,43)(37,40,44,39);(5,7)(6,10)(8,11)(9,12)(13,16,17,24)(14,19,20,26)(15,23,22,28)(18,25,21,27)(29,34,35,42)(30,37,38,44)(31,39,32,40)(33,41,36,43)Polycyclic presentation:
2^9|a^2=g,b^2=fh,c^2=fh,d^2=fgi,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4408106);