Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| 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 | |
1756 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)(5,6,8,10)(7,11,12,9)(15,21)(16,23)(19,24)(20,25)(29,30)(31,32)(33,37)(34,35)(36,40)(38,39)(41,44)(42,43);(5,7,8,12)(6,9,10,11)(14,18)(19,24)(20,25)(26,28)(30,35)(31,36)(37,43)(38,41);(6,10)(9,11)(13,14,17,18)(15,19,21,24)(16,20,23,25)(22,26,27,28)(29,31,34,36)(30,32,35,40)(33,41,42,38)(37,44,43,39);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11)(13,15,17,21)(14,19,18,24)(16,22,23,27)(20,26,25,28)(29,32)(30,36)(31,35)(33,39)(34,40)(37,41)(38,43)(42,44);(5,6)(7,9)(8,10)(11,12)(13,16)(14,20)(15,22)(17,23)(18,25)(19,26)(21,27)(24,28)(29,33,34,42)(30,37,35,43)(31,38,36,41)(32,39,40,44)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=h,d^2=fgh,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4508852);