Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4131229 |
| 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 | |
1781 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)(5,6,8,10)(7,9,11,12)(13,14,17,18)(15,21,23,19)(16,24,25,20)(22,27,26,28)(29,30)(31,37)(32,35)(33,36)(34,39)(38,44)(40,42)(41,43);(7,11)(9,12)(14,18)(16,25)(19,21)(22,26)(29,30,33,36)(31,34,39,37)(32,35,42,40)(38,43,41,44);(5,7,8,11)(6,9,10,12)(13,14)(15,21)(16,24)(17,18)(19,23)(20,25)(22,27)(26,28)(29,30)(31,34)(32,40)(33,36)(35,42)(37,39)(38,44)(41,43);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,15,17,23)(14,19,18,21)(16,26,25,22)(20,28,24,27)(29,31,33,39)(30,34,36,37)(32,41,42,38)(35,44,40,43);(13,16,17,25)(14,20,18,24)(15,22,23,26)(19,27,21,28)(29,32)(30,35)(31,38)(33,42)(34,43)(36,40)(37,44)(39,41)Polycyclic presentation:
2^9|a^2=fgh,b^2=fg,c^2=fh,d^2=f,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4042664);