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 | |
1620 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)(7,11)(8,10)(9,12)(13,14,18,15)(16,19,23,21)(17,22,25,20)(24,27,28,26)(30,34)(32,40)(35,37)(38,44);(5,7)(6,11)(8,9)(10,12)(14,15)(16,23)(20,22)(24,28)(29,30,33,34)(31,37,39,35)(32,41,40,36)(38,42,44,43);(6,8)(7,12)(13,15,18,14)(16,19,23,21)(17,20,25,22)(24,27,28,26)(29,30)(31,35)(32,41)(33,34)(36,40)(37,39)(38,43)(42,44);(1,2)(3,4)(5,8,10,6)(7,9,12,11)(13,16)(14,19)(15,21)(17,24)(18,23)(20,26)(22,27)(25,28)(29,31,33,39)(30,35,34,37)(32,38,40,44)(36,42,41,43);(5,9)(6,7)(8,12)(10,11)(13,17,18,25)(14,20,15,22)(16,24,23,28)(19,26,21,27)(29,32,33,40)(30,36,34,41)(31,38,39,44)(35,42,37,43)Polycyclic presentation:
2^9|a^2=fgi,b^2=fgh,c^2=gh,d^2=g,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, 4950134);