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 | |
1712 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)(7,8)(11,12)(13,14)(15,21)(16,22)(17,20)(18,19)(23,26)(24,28)(25,27)(29,30,34,37)(31,35,32,39)(33,36,41,44)(38,42,40,43);(6,10)(11,12)(14,19)(15,22)(17,25)(24,26)(29,31,34,32)(30,35,37,39)(33,40,41,38)(36,43,44,42);(5,6)(7,11)(8,12)(9,10)(13,15)(14,16)(17,23)(18,22)(19,21)(20,24)(25,28)(26,27)(29,32,34,31)(30,35,37,39)(33,40,41,38)(36,42,44,43);(1,2)(3,4)(5,7,9,8)(6,11,10,12)(13,16)(14,15)(17,26)(18,21)(19,22)(20,28)(23,27)(24,25)(29,30)(31,35)(32,39)(33,36)(34,37)(38,42)(40,43)(41,44);(5,8,9,7)(6,12,10,11)(13,17,18,25)(14,20,19,27)(15,23,22,28)(16,24,21,26)(29,33,34,41)(30,36,37,44)(31,38,32,40)(35,42,39,43)Polycyclic presentation:
2^9|a^2=fg,b^2=fg,c^2=fg,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, 5538274);