Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5035983 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C2×#392 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2068 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)(5,6,10,11)(7,9,8,12)(14,18)(15,21)(20,25)(23,27)(29,30)(31,37)(32,34)(33,36)(35,39)(38,44)(40,41)(42,43);(5,7,10,8)(6,9,11,12)(16,24)(20,25)(23,27)(26,28)(30,32)(33,41)(35,37)(38,42);(6,11)(7,8)(13,14)(15,22)(16,20)(17,18)(19,21)(23,28)(24,25)(26,27)(29,31)(30,35)(32,37)(33,42)(34,39)(36,44)(38,41)(40,43);(1,2,3,4)(5,8,10,7)(6,9,11,12)(13,15)(14,19)(16,23)(17,21)(18,22)(20,26)(24,27)(25,28)(29,32,34,30)(31,37,39,35)(33,40,41,36)(38,44,42,43);(5,9)(6,7)(8,11)(10,12)(13,16,17,24)(14,20,18,25)(15,23,21,27)(19,26,22,28)(29,33,34,41)(30,36,32,40)(31,38,39,42)(35,43,37,44)Polycyclic presentation:
2^9|a^2=h,b^2=h,d^2=fghi,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, 4992617);