Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2601293 |
| 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 | |
1632 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)(13,14,17,18)(15,21,22,19)(16,24,26,20)(23,27,25,28)(29,30,35,32)(31,33,39,36)(34,37,43,40)(38,41,44,42);(6,9)(10,12)(14,18)(15,22)(20,24)(23,25)(29,31,35,39)(30,36,32,33)(34,38,43,44)(37,42,40,41);(5,6)(7,10)(8,9)(11,12)(13,14)(15,19)(16,20)(17,18)(21,22)(23,27)(24,26)(25,28)(29,32,35,30)(31,33,39,36)(34,40,43,37)(38,41,44,42);(1,2)(3,4)(13,15,17,22)(14,19,18,21)(16,25,26,23)(20,28,24,27)(29,33)(30,31)(32,39)(34,42)(35,36)(37,44)(38,40)(41,43);(5,7,8,11)(6,10,9,12)(13,16)(14,20)(15,23)(17,26)(18,24)(19,27)(21,28)(22,25)(29,34)(30,37)(31,38)(32,40)(33,41)(35,43)(36,42)(39,44)Polycyclic presentation:
2^9|a^2=i,b^2=fg,c^2=fg,d^2=g,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2520783);