Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4379698 |
| 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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1738 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6,9,12)(7,10,8,11)(13,14)(15,19)(16,18)(17,20)(21,23)(22,27)(24,26)(25,28)(29,30,33,34)(31,37,38,35)(32,40,42,36)(39,43,41,44);(5,7,9,8)(6,10,12,11)(14,16)(19,21)(20,24)(27,28)(30,34)(31,38)(36,40)(39,41);(5,6)(7,10)(8,11)(9,12)(13,15,18,23)(14,19,16,21)(17,25,26,22)(20,28,24,27)(29,31,33,38)(30,35,34,37)(32,41,42,39)(36,44,40,43);(1,2)(3,4)(5,6)(7,10)(8,11)(9,12)(13,16,18,14)(15,21,23,19)(17,24,26,20)(22,28,25,27)(29,30)(31,35)(32,36)(33,34)(37,38)(39,43)(40,42)(41,44);(5,8,9,7)(6,11,12,10)(13,17,18,26)(14,20,16,24)(15,22,23,25)(19,27,21,28)(29,32)(30,36)(31,39)(33,42)(34,40)(35,43)(37,44)(38,41)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=f,d^2=fh,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4288458);