Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499897 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#493 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #376 ◁ #1726 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
703 conjugacy classes of subgroups; only the 152 normal subgroups are shown.
This group has 703 conjugacy classes of subgroups — too many to draw. Only its 152 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)(7,11)(8,9)(10,12)(13,14,18,23)(15,19,24,32)(16,20,29,36)(17,30,28,22)(21,37,35,27)(25,31,38,42)(26,41,39,34)(33,44,43,40);(1,2,3,4)(6,9)(10,11)(13,15)(14,19)(16,25)(17,26)(18,24)(20,31)(21,33)(22,34)(23,32)(27,40)(28,39)(29,38)(30,41)(35,43)(36,42)(37,44);(2,4)(14,20)(16,27)(17,28)(19,31)(21,22)(23,36)(25,40)(26,39)(29,37)(30,35)(32,42)(33,34)(38,44)(41,43);(5,7,8,12)(6,10,9,11)(15,24)(19,32)(25,38)(26,39)(31,42)(33,43)(34,41)(40,44)Polycyclic presentation:
2^9|a^2=i,b^2=g,d^2=e,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=cg,d^b=di,f^c=fh,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 413478);