Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| 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 | |
1663 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)(5,6,10,9)(7,12,8,11)(14,18)(15,21)(20,25)(23,27)(29,30,33,34)(31,37,39,35)(32,40,41,36)(38,42,44,43);(5,7,10,8)(6,11,9,12)(16,24)(20,25)(23,27)(26,28)(30,34)(35,37)(36,40)(42,43);(7,8)(11,12)(13,14,17,18)(15,22,21,19)(16,20,24,25)(23,28,27,26)(29,31)(30,35)(32,38)(33,39)(34,37)(36,42)(40,43)(41,44);(1,2)(3,4)(5,8,10,7)(6,11,9,12)(13,15,17,21)(14,19,18,22)(16,23,24,27)(20,26,25,28)(29,32,33,41)(30,36,34,40)(31,38,39,44)(35,42,37,43);(5,9,10,6)(7,12,8,11)(13,16)(14,20)(15,23)(17,24)(18,25)(19,26)(21,27)(22,28)(29,30)(31,35)(32,40)(33,34)(36,41)(37,39)(38,43)(42,44)Polycyclic presentation:
2^9|a^2=f,b^2=h,c^2=fgh,d^2=gh,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5544040);