Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #93784 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1347 |
| commutator | C22×C16 | C2×C4 |
| Frattini, cosocle | C2×C4⋊C16 | C22 |
| prime radical | C23 | #357 |
| LCS | G ▷ C22×C16 ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ C2×C4⋊C16 ◁ G | |
| LECS | G ▷ C2×C4⋊C16 ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
142 conjugacy classes of subgroups; only the 32 normal subgroups are shown.
This group has 142 conjugacy classes of subgroups — too many to draw. Only its 32 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 72 points:
(1,2,3,6,4,5,7,8)(9,10)(11,24,26,55)(12,31,34,54)(13,39)(14,42)(15,22)(16,21)(17,23)(18,25)(19,35,49,36)(20,33)(27,62,28,63)(29,57,30,56)(32,52)(37,65,38,64)(40,71,43,48)(41,53)(44,67)(45,60)(46,59)(47,61)(50,69,51,68)(58,72)(66,70);(2,5)(9,11,17,26)(10,18,23,47)(12,32,34,58)(13,40,14,43)(15,30,16,29)(19,48,49,71)(20,52,53,72)(21,46,22,45)(24,54,55,31)(25,35,61,36)(27,33,28,41)(37,67,38,66)(39,70,42,44)(50,63,51,62)(56,64,57,65)(59,69,60,68)Polycyclic presentation:
2^9|a^2=d,b^2=i,c^2=fh,d^2=gh,e^2=hi,f^2=hi,g^2=i,h^2=i,b^a=bc,c^a=ce,e^a=eg,f^a=fh,g^a=gi,h^a=hi,c^b=cf,d^b=defh,e^b=eh,f^b=fh,g^b=gi,h^b=hi,d^c=dghi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1045);