Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #498791 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #376 ◁ #1726 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
816 conjugacy classes of subgroups; only the 170 normal subgroups are shown.
This group has 816 conjugacy classes of subgroups — too many to draw. Only its 170 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) = 256 |
A minimal faithful permutation representation — 4 generators on 44 points:
(5,6,8,9)(7,10,11,12)(13,14,16,21,30,38,43,36,18,23,31,35,17,22,29,20)(15,24,26,39,32,44,42,33,25,40,41,34,27,37,28,19);(1,2)(3,4)(6,9)(10,12)(13,15)(14,19)(16,28)(17,32)(18,25)(20,24)(21,37)(22,39)(23,33)(26,29)(27,30)(31,42)(34,38)(35,44)(36,40)(41,43);(2,4)(14,20)(16,29)(17,30)(19,24)(21,22)(23,36)(26,28)(27,32)(31,43)(33,40)(34,44)(35,38)(37,39)(41,42);(5,7,8,11)(6,10,9,12)(15,25)(19,33)(24,40)(26,41)(27,32)(28,42)(34,39)(37,44)Polycyclic presentation:
2^9|a^2=e,d^2=efh,e^2=hi,f^2=hi,h^2=i,b^a=be,c^a=cf,c^b=cg,d^b=di,e^b=eh,f^b=fh,h^b=hi,e^c=eh,f^c=fh,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406843);