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 | |
1632 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)(5,6,8,11)(7,9,12,10)(15,21)(16,23)(19,24)(20,25)(29,30)(31,40)(32,36)(33,42)(34,38)(35,39)(37,43)(41,44);(5,7)(6,9)(8,12)(10,11)(14,18)(19,24)(20,25)(26,28)(29,31,35,32)(30,36,39,40)(33,38,43,41)(34,42,44,37);(5,7,8,12)(6,10,11,9)(13,14)(15,19)(16,20)(17,18)(21,24)(22,26)(23,25)(27,28)(29,32,35,31)(30,36,39,40)(33,41,43,38)(34,42,44,37);(1,2,3,4)(5,7)(6,10)(8,12)(9,11)(13,15)(14,19)(16,22)(17,21)(18,24)(20,26)(23,27)(25,28)(29,33,35,43)(30,37,39,42)(31,41,32,38)(34,36,44,40);(5,6)(7,10)(8,11)(9,12)(13,16,17,23)(14,20,18,25)(15,22,21,27)(19,26,24,28)(29,34,35,44)(30,38,39,41)(31,42,32,37)(33,36,43,40)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=fg,d^2=hi,e^2=fgh,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, 4991500);