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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1705 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(13,14,17,18)(15,21,22,19)(16,24,26,20)(23,27,25,28)(29,30,34,38)(31,32,36,35)(33,41,42,37)(39,43,44,40);(5,6,8,10)(7,9,11,12)(14,18)(15,22)(20,24)(23,25)(31,36)(32,35)(39,44)(40,43);(5,6)(7,9)(8,10)(11,12)(13,14)(15,19)(16,20)(17,18)(21,22)(23,27)(24,26)(25,28)(29,31,34,36)(30,35,38,32)(33,39,42,44)(37,43,41,40);(1,2)(3,4)(13,15,17,22)(14,19,18,21)(16,25,26,23)(20,28,24,27)(29,32,34,35)(30,36,38,31)(33,40,42,43)(37,44,41,39);(5,7,8,11)(6,9,10,12)(13,16,17,26)(14,20,18,24)(15,23,22,25)(19,27,21,28)(29,33,34,42)(30,37,38,41)(31,39,36,44)(32,40,35,43)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=fg,d^2=g,e^2=h,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, 2513133);