Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4417862 |
| 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 | |
1666 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)(6,7)(10,11)(13,14,18,19)(15,17,21,27)(16,23,25,20)(22,24,28,26)(29,30,35,32)(31,37,33,36)(34,38,42,40)(39,44,41,43);(14,19)(15,21)(20,23)(22,28)(29,31,35,33)(30,36,32,37)(34,39,42,41)(38,43,40,44);(5,6,9,7)(8,10,12,11)(13,15)(14,17)(16,22)(18,21)(19,27)(20,26)(23,24)(25,28)(29,32,35,30)(31,37,33,36)(34,40,42,38)(39,44,41,43);(1,2,3,4)(5,7,9,6)(8,11,12,10)(13,16,18,25)(14,20,19,23)(15,22,21,28)(17,26,27,24)(29,33,35,31)(30,37,32,36)(34,39,42,41)(38,43,40,44);(5,8,9,12)(6,10,7,11)(13,17)(14,15)(16,24)(18,27)(19,21)(20,28)(22,23)(25,26)(29,34)(30,38)(31,39)(32,40)(33,41)(35,42)(36,43)(37,44)Polycyclic presentation:
2^9|a^2=f,b^2=fh,c^2=fgh,d^2=i,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4341949);