Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5473759 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
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 | |
1832 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)(5,6,9,7)(8,11,12,10)(13,14,18,16)(15,21,23,19)(17,20,25,24)(22,27,28,26)(29,30,34,35)(31,36,38,33)(32,40,42,37)(39,41,44,43);(5,6,9,7)(8,11,12,10)(17,25)(20,24)(22,28)(26,27)(30,35)(33,36)(37,40)(41,43);(6,7)(10,11)(13,15,18,23)(14,19,16,21)(17,22,25,28)(20,26,24,27)(29,31)(30,36)(32,39)(33,35)(34,38)(37,43)(40,41)(42,44);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,16,18,14)(15,19,23,21)(17,24,25,20)(22,26,28,27)(29,32,34,42)(30,37,35,40)(31,39,38,44)(33,41,36,43);(5,8)(6,10)(7,11)(9,12)(13,17)(14,20)(15,22)(16,24)(18,25)(19,26)(21,27)(23,28)(29,33,34,36)(30,38,35,31)(32,41,42,43)(37,44,40,39)Polycyclic presentation:
2^9|a^2=fg,b^2=h,c^2=gh,d^2=fg,e^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, 5382802);