Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4932388 |
| 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, prime radical, cosocle | C24 | C25 |
| 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) = 512 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,3,4)(5,6,9,10)(7,8,12,11)(13,14,17,18)(15,21,23,19)(16,24,25,20)(22,26,28,27)(30,34)(35,41)(36,39)(42,43);(14,18)(19,21)(20,24)(26,27)(29,30,33,34)(31,35,38,41)(32,39,40,36)(37,43,44,42);(6,10)(7,12)(13,15,17,23)(14,19,18,21)(16,22,25,28)(20,26,24,27)(29,30)(31,35)(32,36)(33,34)(37,42)(38,41)(39,40)(43,44);(1,2)(3,4)(5,7,9,12)(6,11,10,8)(13,16)(14,20)(15,22)(17,25)(18,24)(19,26)(21,27)(23,28)(29,31,33,38)(30,35,34,41)(32,37,40,44)(36,42,39,43);(5,8,9,11)(6,12,10,7)(13,14)(15,19)(16,20)(17,18)(21,23)(22,26)(24,25)(27,28)(29,32)(30,36)(31,37)(33,40)(34,39)(35,42)(38,44)(41,43)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=fh,d^2=fgh,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=ef,d^c=dfg80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4851790);