Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, prime radical, cosocle | C24 | C25 |
| commutator | C23 | C24×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1834 conjugacy classes of subgroups; only the 794 normal subgroups are shown.
This group has 1834 conjugacy classes of subgroups — too many to draw. Only its 794 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,3,4)(7,11)(10,12)(13,14,18,15)(16,21,24,19)(17,22,25,20)(23,26,28,27)(30,34)(31,37)(32,40)(42,44);(6,9)(10,12)(14,15)(19,21)(20,22)(26,27)(29,30,33,34)(31,35,37,38)(32,36,40,41)(39,42,43,44);(5,6,8,9)(7,10,11,12)(13,15,18,14)(16,19,24,21)(17,22,25,20)(23,26,28,27)(29,31)(30,35)(32,39)(33,37)(34,38)(36,42)(40,43)(41,44);(5,7)(6,10)(8,11)(9,12)(13,16,18,24)(14,19,15,21)(17,23,25,28)(20,26,22,27)(29,30)(31,38)(32,36)(33,34)(35,37)(39,44)(40,41)(42,43);(13,17,18,25)(14,20,15,22)(16,23,24,28)(19,26,21,27)(29,32,33,40)(30,36,34,41)(31,39,37,43)(35,42,38,44)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=h,d^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=dgh,e^a=ef,c^b=ch,d^b=dg,d^c=df92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 489425);