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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1828 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)(5,6,9,10)(7,11,12,8)(14,18)(15,21)(20,25)(23,27)(29,30,33,34)(31,37,38,35)(32,40,41,36)(39,42,44,43);(5,7,9,12)(6,8,10,11)(16,24)(20,25)(23,27)(26,28)(30,34)(31,38)(32,41)(42,43);(6,10)(8,11)(13,14)(15,22)(16,20)(17,18)(19,21)(23,28)(24,25)(26,27)(29,30)(31,35)(32,40)(33,34)(36,41)(37,38)(39,43)(42,44);(1,2,3,4)(5,7)(6,8)(9,12)(10,11)(13,15)(14,19)(16,23)(17,21)(18,22)(20,26)(24,27)(25,28)(29,31,33,38)(30,35,34,37)(32,39,41,44)(36,42,40,43);(5,8)(6,7)(9,11)(10,12)(13,16,17,24)(14,20,18,25)(15,23,21,27)(19,26,22,28)(29,32,33,41)(30,36,34,40)(31,39,38,44)(35,42,37,43)Polycyclic presentation:
2^9|a^2=f,b^2=fh,d^2=ghi,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, 4992094);