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 | |
1821 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)(7,11)(9,12)(13,14)(15,18)(16,21)(17,22)(19,23)(20,25)(24,26)(27,28)(29,30,34,35)(31,32,36,40)(33,41,42,37)(38,44,43,39);(5,6,8,10)(7,9,11,12)(14,15)(16,23)(17,25)(26,27)(30,35)(31,36)(33,42)(39,44);(5,6)(7,9)(8,10)(11,12)(13,15,18,14)(16,19,23,21)(17,22,25,20)(24,26,28,27)(29,31)(30,32)(33,38)(34,36)(35,40)(37,39)(41,44)(42,43);(1,2)(3,4)(13,16,18,23)(14,19,15,21)(17,24,25,28)(20,26,22,27)(29,32)(30,36)(31,35)(33,39)(34,40)(37,43)(38,41)(42,44);(5,7)(6,9)(8,11)(10,12)(13,17)(14,20)(15,22)(16,24)(18,25)(19,26)(21,27)(23,28)(29,33,34,42)(30,37,35,41)(31,38,36,43)(32,39,40,44)Polycyclic presentation:
2^9|a^2=fg,b^2=fh,c^2=g,d^2=g,e^2=fg,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, 4374047);