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 | |
1724 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)(6,7)(8,12)(13,14,17,20)(15,21,22,18)(16,24,25,19)(23,26,28,27)(29,30)(31,36)(32,37)(33,38)(34,35)(39,44)(40,42)(41,43);(5,6,9,7)(8,11,12,10)(15,22)(16,25)(18,21)(19,24)(30,35)(31,37)(33,41)(40,44);(8,12)(10,11)(13,15)(14,18)(16,23)(17,22)(19,26)(20,21)(24,27)(25,28)(29,31,34,37)(30,36,35,32)(33,42,41,39)(38,40,43,44);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,14)(15,21)(16,19)(17,20)(18,22)(23,27)(24,25)(26,28)(29,32,34,36)(30,37,35,31)(33,40,41,44)(38,42,43,39);(5,8)(6,10)(7,11)(9,12)(13,16)(14,19)(15,23)(17,25)(18,26)(20,24)(21,27)(22,28)(29,33)(30,38)(31,39)(32,40)(34,41)(35,43)(36,44)(37,42)Polycyclic presentation:
2^9|a^2=g,b^2=fgh,c^2=gh,d^2=f,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, 5382696);