Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| 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 | |
1718 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)(5,6,9,8)(7,10,12,11)(14,18)(15,21)(16,24)(26,28)(29,30)(31,37)(32,36)(33,34)(35,38)(39,43)(40,41)(42,44);(6,8)(10,11)(14,18)(16,24)(19,22)(23,27)(30,34)(31,38)(36,40)(39,44);(5,7,9,12)(6,10,8,11)(13,14)(15,22)(16,20)(17,18)(19,21)(23,28)(24,25)(26,27)(29,30)(31,37)(32,40)(33,34)(35,38)(36,41)(39,42)(43,44);(1,2)(3,4)(5,8,9,6)(7,10,12,11)(13,15,17,21)(14,19,18,22)(16,23,24,27)(20,26,25,28)(29,31)(30,35)(32,39)(33,38)(34,37)(36,42)(40,43)(41,44);(5,7)(6,11)(8,10)(9,12)(13,16,17,24)(14,20,18,25)(15,23,21,27)(19,26,22,28)(29,32,33,41)(30,36,34,40)(31,39,38,44)(35,42,37,43)Polycyclic presentation:
2^9|a^2=fgh,c^2=fgh,d^2=fh,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5134200);