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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1726 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6,10,11)(7,12,8,9)(13,14)(15,21)(16,18)(17,24)(19,23)(20,26)(22,27)(25,28)(29,30)(31,39)(32,37)(33,38)(34,35)(36,41)(40,43)(42,44);(6,11)(7,8)(14,16)(19,21)(20,24)(27,28)(29,31,35,41)(30,36,34,39)(32,40,42,33)(37,38,44,43);(5,7,10,8)(6,9,11,12)(13,15,18,23)(14,19,16,21)(17,25,26,22)(20,28,24,27)(29,32)(30,37)(31,40)(33,41)(34,44)(35,42)(36,38)(39,43);(1,2,3,4)(5,8,10,7)(6,9,11,12)(13,16,18,14)(15,21,23,19)(17,24,26,20)(22,28,25,27)(29,33,35,40)(30,38,34,43)(31,32,41,42)(36,44,39,37);(5,9)(6,7)(8,11)(10,12)(13,17,18,26)(14,20,16,24)(15,22,23,25)(19,27,21,28)(29,34,35,30)(31,39,41,36)(32,37,42,44)(33,43,40,38)Polycyclic presentation:
2^9|a^2=h,b^2=fg,c^2=g,d^2=fghi,e^2=fh,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, 4579318);