Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5473759 |
| 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 | |
1754 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)(5,6,10,8)(7,12,9,11)(13,14,17,20)(15,18,21,25)(16,23,24,19)(22,28,27,26)(29,30,34,35)(31,36,38,32)(33,37,41,43)(39,44,42,40);(5,6,10,8)(7,11,9,12)(15,21)(18,25)(22,27)(26,28)(30,35)(31,38)(33,41)(40,44);(5,7,10,9)(6,11,8,12)(13,15)(14,18)(16,22)(17,21)(19,26)(20,25)(23,28)(24,27)(29,31)(30,36)(32,35)(33,42)(34,38)(37,40)(39,41)(43,44);(1,2)(3,4)(5,8,10,6)(7,11,9,12)(13,14)(15,18)(16,19)(17,20)(21,25)(22,26)(23,24)(27,28)(29,32,34,36)(30,31,35,38)(33,40,41,44)(37,39,43,42);(5,9,10,7)(6,11,8,12)(13,16,17,24)(14,19,20,23)(15,22,21,27)(18,26,25,28)(29,33)(30,37)(31,39)(32,40)(34,41)(35,43)(36,44)(38,42)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=g,d^2=fg,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5387219);