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 | |
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:
(2,4)(5,6,9,7)(8,10,12,11)(13,14)(15,18)(16,21)(17,22)(19,23)(20,25)(24,26)(27,28)(30,32)(33,41)(35,37)(38,44);(6,7)(8,12)(14,15)(16,23)(17,25)(26,27)(29,30,34,32)(31,37,39,35)(33,40,41,36)(38,42,44,43);(6,7)(10,11)(13,15,18,14)(16,19,23,21)(17,22,25,20)(24,26,28,27)(29,31,34,39)(30,35,32,37)(33,38,41,44)(36,42,40,43);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,16,18,23)(14,19,15,21)(17,24,25,28)(20,26,22,27)(29,32,34,30)(31,35,39,37)(33,40,41,36)(38,42,44,43);(5,8,9,12)(6,10,7,11)(13,17)(14,20)(15,22)(16,24)(18,25)(19,26)(21,27)(23,28)(29,33,34,41)(30,36,32,40)(31,38,39,44)(35,42,37,43)Polycyclic presentation:
2^9|a^2=fh,b^2=fg,c^2=h,d^2=f,e^2=gh,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, 4380951);