Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5035983 |
| automorphism | Aut(G) order 2097152 |
| 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 | |
1856 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)(5,6,8,9)(7,11,12,10)(15,21)(16,23)(19,24)(20,25)(29,30)(31,37)(32,34)(33,36)(35,39)(38,44)(40,41)(42,43);(6,9)(10,11)(14,18)(19,24)(20,25)(26,28)(30,32)(33,41)(35,37)(38,42);(5,6,8,9)(7,11,12,10)(13,14)(15,19)(16,20)(17,18)(21,24)(22,26)(23,25)(27,28)(29,31)(30,35)(32,37)(33,42)(34,39)(36,44)(38,41)(40,43);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,19)(16,22)(17,21)(18,24)(20,26)(23,27)(25,28)(29,32,34,30)(31,37,39,35)(33,40,41,36)(38,44,42,43);(5,7)(6,10)(8,12)(9,11)(13,16,17,23)(14,20,18,25)(15,22,21,27)(19,26,24,28)(29,33,34,41)(30,36,32,40)(31,38,39,42)(35,43,37,44)Polycyclic presentation:
2^9|a^2=fg,c^2=fg,d^2=hi,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4991438);