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 | |
1598 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 44 points:
(6,9)(10,12)(14,18)(15,21)(16,23)(26,28)(29,30,34,32)(31,37,38,35)(33,40,41,36)(39,42,44,43);(1,2,3,4)(14,18)(15,21)(20,25)(22,27)(30,32)(31,38)(36,40)(39,44);(1,2)(3,4)(13,14)(15,19)(16,20)(17,18)(21,24)(22,26)(23,25)(27,28)(29,31,34,38)(30,35,32,37)(33,39,41,44)(36,42,40,43);(5,6)(7,10)(8,9)(11,12)(13,15,17,21)(14,19,18,24)(16,22,23,27)(20,26,25,28)(29,32,34,30)(31,37,38,35)(33,36,41,40)(39,42,44,43);(5,7,8,11)(6,10,9,12)(13,16,17,23)(14,20,18,25)(15,22,21,27)(19,26,24,28)(29,33,34,41)(30,36,32,40)(31,39,38,44)(35,42,37,43)Polycyclic presentation:
2^9|a^2=f,b^2=gh,c^2=f,d^2=g,e^2=fi,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, 2579998);