Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4861280 |
| 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 | |
1708 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)(6,9)(7,11)(13,14)(15,21)(16,20)(17,18)(19,23)(22,27)(24,25)(26,28)(29,30,34,35)(31,32,37,40)(33,41,43,36)(38,42,44,39);(5,6,8,9)(7,10,11,12)(14,18)(19,21)(20,24)(26,27)(30,35)(31,37)(36,41)(38,44);(5,6)(7,10)(8,9)(11,12)(13,15,17,23)(14,19,18,21)(16,22,25,28)(20,26,24,27)(29,31)(30,32)(33,38)(34,37)(35,40)(36,39)(41,42)(43,44);(1,2)(3,4)(5,6)(7,12)(8,9)(10,11)(13,14)(15,19)(16,24)(17,18)(20,25)(21,23)(22,27)(26,28)(29,32)(30,31)(33,42)(34,40)(35,37)(36,44)(38,41)(39,43);(5,7)(6,10)(8,11)(9,12)(13,16)(14,20)(15,22)(17,25)(18,24)(19,26)(21,27)(23,28)(29,33,34,43)(30,36,35,41)(31,38,37,44)(32,39,40,42)Polycyclic presentation:
2^9|a^2=fg,b^2=fgh,c^2=fh,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4770547);