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 | |
1692 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)(7,12)(8,9)(10,11)(15,21)(16,23)(19,24)(20,25)(29,30,34,37)(31,35,32,38)(33,36,41,43)(39,42,40,44);(5,7)(6,9)(8,10)(11,12)(14,18)(19,24)(20,25)(26,28)(29,31,34,32)(30,35,37,38)(33,39,41,40)(36,42,43,44);(6,11)(7,8)(13,14)(15,19)(16,20)(17,18)(21,24)(22,26)(23,25)(27,28)(29,32,34,31)(30,35,37,38)(33,40,41,39)(36,42,43,44);(1,2,3,4)(5,8,10,7)(6,12,11,9)(13,15)(14,19)(16,22)(17,21)(18,24)(20,26)(23,27)(25,28)(29,30)(31,38)(32,35)(33,36)(34,37)(39,44)(40,42)(41,43);(5,9,10,12)(6,7,11,8)(13,16,17,23)(14,20,18,25)(15,22,21,27)(19,26,24,28)(29,33,34,41)(30,36,37,43)(31,39,32,40)(35,42,38,44)Polycyclic presentation:
2^9|a^2=fg,b^2=fg,c^2=fg,d^2=ghi,e^2=fgh,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, 4410259);