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 | |
1773 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)(7,11)(8,10)(9,12)(13,14)(15,16)(17,20)(18,19)(21,24)(22,23)(25,28)(26,27)(29,30,34,37)(31,38,32,35)(33,41,42,36)(39,43,40,44);(5,7)(6,11)(8,9)(10,12)(14,19)(15,21)(20,27)(22,28)(29,31,34,32)(30,35,37,38)(33,39,42,40)(36,43,41,44);(6,8)(7,12)(13,15,18,21)(14,16,19,24)(17,22,26,28)(20,23,27,25)(29,32,34,31)(30,35,37,38)(33,40,42,39)(36,43,41,44);(1,2)(3,4)(5,8,10,6)(7,9,12,11)(13,16)(14,15)(17,25)(18,24)(19,21)(20,28)(22,27)(23,26)(29,30)(31,35)(32,38)(33,41)(34,37)(36,42)(39,44)(40,43);(5,9)(6,7)(8,12)(10,11)(13,17)(14,20)(15,22)(16,23)(18,26)(19,27)(21,28)(24,25)(29,33,34,42)(30,36,37,41)(31,39,32,40)(35,43,38,44)Polycyclic presentation:
2^9|a^2=fgh,b^2=fgh,c^2=f,d^2=g,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, 5533274);