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 | |
1758 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)(7,9)(8,10)(11,12)(13,14,18,19)(15,23,26,20)(16,25,27,21)(17,24,28,22)(29,30)(31,37)(32,35)(33,36)(34,39)(38,43)(40,41)(42,44);(6,7)(10,11)(14,19)(16,27)(20,23)(22,24)(29,30,33,36)(31,34,39,37)(32,35,41,40)(38,42,44,43);(5,6,9,7)(8,11,12,10)(13,15,18,26)(14,20,19,23)(16,24,27,22)(17,21,28,25)(29,30)(31,34)(32,40)(33,36)(35,41)(37,39)(38,43)(42,44);(1,2,3,4)(5,7,9,6)(8,11,12,10)(13,16,18,27)(14,21,19,25)(15,24,26,22)(17,20,28,23)(29,31)(30,34)(32,38)(33,39)(35,42)(36,37)(40,43)(41,44);(5,8)(6,10)(7,11)(9,12)(13,17,18,28)(14,22,19,24)(15,25,26,21)(16,20,27,23)(29,32)(30,35)(31,38)(33,41)(34,42)(36,40)(37,43)(39,44)Polycyclic presentation:
2^9|a^2=fh,b^2=fg,c^2=g,d^2=fgi,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, 4577271);