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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1781 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)(8,10,12,11)(13,14)(15,22)(16,17)(18,19)(20,25)(21,26)(23,28)(24,27)(29,30)(31,34)(32,40)(33,37)(35,39)(36,42)(38,44)(41,43);(8,12)(10,11)(14,19)(17,26)(20,22)(24,28)(29,31,33,39)(30,34,37,35)(32,41,42,38)(36,44,40,43);(6,7)(10,11)(13,15,18,25)(14,20,19,22)(16,23,21,27)(17,24,26,28)(29,31)(30,35)(32,38)(33,39)(34,37)(36,44)(40,43)(41,42);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,16)(14,17)(15,23)(18,21)(19,26)(20,24)(22,28)(25,27)(29,30)(31,34)(32,36)(33,37)(35,39)(38,43)(40,42)(41,44);(5,8)(6,10)(7,11)(9,12)(13,17)(14,21)(15,24)(16,19)(18,26)(20,27)(22,23)(25,28)(29,32)(30,36)(31,38)(33,42)(34,43)(35,44)(37,40)(39,41)Polycyclic presentation:
2^9|a^2=fgh,b^2=h,c^2=f,d^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, 5524496);