Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5035983 |
| 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 | |
1626 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)(5,6,10,11)(7,9,8,12)(14,18)(19,21)(20,25)(26,27)(29,30)(31,34)(32,35)(33,38)(36,41)(37,39)(40,43)(42,44);(5,7,10,8)(6,9,11,12)(16,24)(20,25)(22,28)(26,27)(30,31)(32,39)(33,41)(42,43);(6,11)(7,8)(13,14,17,18)(15,21,23,19)(16,20,24,25)(22,27,28,26)(29,31,34,30)(32,37,39,35)(33,36,41,38)(40,42,44,43);(1,2)(3,4)(5,8,10,7)(6,9,11,12)(13,15,17,23)(14,19,18,21)(16,22,24,28)(20,26,25,27)(29,32)(30,35)(31,37)(33,40)(34,39)(36,42)(38,43)(41,44);(5,9)(6,7)(8,11)(10,12)(13,16)(14,20)(15,22)(17,24)(18,25)(19,26)(21,27)(23,28)(29,33,34,41)(30,36,31,38)(32,40,39,44)(35,42,37,43)Polycyclic presentation:
2^9|a^2=i,b^2=fh,c^2=fgh,d^2=fg,e^2=f,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, 4950097);