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 | |
1681 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)(13,14,17,20)(15,18,21,25)(16,23,24,19)(22,28,27,26)(29,30)(31,35)(32,34)(33,40)(36,39)(37,41)(38,43)(42,44);(5,7)(6,9)(8,10)(11,12)(15,21)(18,25)(22,27)(26,28)(29,31,34,39)(30,35,32,36)(33,38,41,44)(37,42,40,43);(6,11)(7,8)(13,15,17,21)(14,18,20,25)(16,22,24,27)(19,26,23,28)(29,31)(30,36)(32,35)(33,38)(34,39)(37,43)(40,42)(41,44);(1,2)(3,4)(5,8,10,7)(6,12,11,9)(13,14)(15,18)(16,19)(17,20)(21,25)(22,26)(23,24)(27,28)(29,32,34,30)(31,35,39,36)(33,40,41,37)(38,42,44,43);(5,9,10,12)(6,7,11,8)(13,16,17,24)(14,19,20,23)(15,22,21,27)(18,26,25,28)(29,33)(30,37)(31,38)(32,40)(34,41)(35,42)(36,43)(39,44)Polycyclic presentation:
2^9|a^2=gh,b^2=fg,c^2=gh,d^2=g,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, 4381374);