Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4131229 |
| 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 | |
1710 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)(5,6,8,10)(7,9,11,12)(14,18)(15,21)(16,24)(26,28)(30,31)(32,39)(33,41)(43,44);(7,11)(9,12)(14,18)(16,24)(19,22)(23,27)(29,30,34,31)(32,35,39,37)(33,38,41,36)(40,44,42,43);(5,7,8,11)(6,9,10,12)(13,14)(15,22)(16,20)(17,18)(19,21)(23,28)(24,25)(26,27)(29,31,34,30)(32,35,39,37)(33,36,41,38)(40,44,42,43);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,15,17,21)(14,19,18,22)(16,23,24,27)(20,26,25,28)(29,32,34,39)(30,35,31,37)(33,42,41,40)(36,44,38,43);(13,16,17,24)(14,20,18,25)(15,23,21,27)(19,26,22,28)(29,33,34,41)(30,36,31,38)(32,40,39,42)(35,43,37,44)Polycyclic presentation:
2^9|a^2=fgh,b^2=f,c^2=gh,d^2=g,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4042972);