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 | |
1698 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)(7,11,9,12)(13,14,17,20)(15,21,23,18)(16,19,24,26)(22,27,28,25)(29,30,33,34)(31,37,39,35)(32,40,41,36)(38,42,44,43);(5,7,10,9)(6,11,8,12)(16,24)(19,26)(22,28)(25,27)(30,34)(35,37)(36,40)(42,43);(5,8,10,6)(7,11,9,12)(13,15,17,23)(14,18,20,21)(16,22,24,28)(19,25,26,27)(29,31)(30,35)(32,38)(33,39)(34,37)(36,42)(40,43)(41,44);(1,2)(3,4)(5,7,10,9)(6,12,8,11)(13,14)(15,21)(16,19)(17,20)(18,23)(22,27)(24,26)(25,28)(29,32,33,41)(30,36,34,40)(31,38,39,44)(35,42,37,43);(5,9,10,7)(6,11,8,12)(13,16,17,24)(14,19,20,26)(15,22,23,28)(18,25,21,27)(29,30)(31,35)(32,40)(33,34)(36,41)(37,39)(38,43)(42,44)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=fgh,d^2=fh,e^2=fgh,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, 5545375);