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 | |
1838 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)(6,9)(7,11)(13,14,17,18)(15,19,23,21)(16,20,25,26)(22,27,24,28)(29,30)(31,37)(32,34)(33,36)(35,39)(38,44)(40,41)(42,43);(5,6,8,9)(7,10,11,12)(14,18)(19,21)(20,26)(27,28)(30,32)(33,41)(35,37)(38,42);(6,9)(7,11)(13,15,17,23)(14,19,18,21)(16,24,25,22)(20,28,26,27)(29,31)(30,35)(32,37)(33,42)(34,39)(36,44)(38,41)(40,43);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,21)(16,20)(17,18)(19,23)(22,28)(24,27)(25,26)(29,32,34,30)(31,37,39,35)(33,40,41,36)(38,44,42,43);(5,7,8,11)(6,10,9,12)(13,16,17,25)(14,20,18,26)(15,22,23,24)(19,27,21,28)(29,33,34,41)(30,36,32,40)(31,38,39,42)(35,43,37,44)Polycyclic presentation:
2^9|a^2=fgh,b^2=h,c^2=fgh,d^2=fgi,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, 4989846);