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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1696 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)(7,8)(9,11)(10,12)(13,14,17,20)(15,21,22,18)(16,24,25,19)(23,26,28,27)(29,30,34,33)(31,37,39,35)(32,40,41,36)(38,42,44,43);(5,7,9,12)(6,10,11,8)(15,22)(16,25)(18,21)(19,24)(30,33)(32,41)(35,37)(38,44);(5,7,9,12)(6,8,11,10)(13,15)(14,18)(16,23)(17,22)(19,26)(20,21)(24,27)(25,28)(29,31,34,39)(30,35,33,37)(32,38,41,44)(36,42,40,43);(1,2,3,4)(5,7)(6,8)(9,12)(10,11)(13,14)(15,21)(16,19)(17,20)(18,22)(23,27)(24,25)(26,28)(29,32,34,41)(30,36,33,40)(31,38,39,44)(35,42,37,43);(5,8)(6,7)(9,10)(11,12)(13,16)(14,19)(15,23)(17,25)(18,26)(20,24)(21,27)(22,28)(29,33,34,30)(31,35,39,37)(32,40,41,36)(38,42,44,43)Polycyclic presentation:
2^9|a^2=g,b^2=fgh,c^2=f,d^2=hi,e^2=gh,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, 4992009);