Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7349 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1317 |
| commutator | C24 | C22×C8 |
| Frattini, cosocle | #407 | C23 |
| socle | C2 | C2×#1317 |
| prime radical | C25 | C2×C8 |
| LCS | G ▷ C24 ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C23×C4 ◁ C2×#407 ◁ G | |
| LECS | G ▷ #407 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
436 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 436 conjugacy classes of subgroups — too many to draw. Only its 80 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,11,6,12,19,26,7,10,18,25,4,9,16,24)(3,8,15,22,13,21,28,31,14,23,17,30,20,27,29,32);(1,3)(2,8,10,23)(4,13)(5,17)(6,20)(7,14)(9,21,12,27)(11,22,25,30)(15,18)(16,29)(19,28)(24,31,26,32);(1,4,7,6)(2,9,10,12)(3,13,14,20)(5,16,18,19)(8,21,23,27)(11,24,25,26)(15,28,17,29)(22,31,30,32)Polycyclic presentation:
2^9|a^2=e,b^2=h,c^2=i,e^2=g,g^2=i,b^a=bd,d^a=df,f^a=fh,h^a=hi,d^b=di,e^b=ef,g^b=gi,e^d=eh,f^e=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72339712, #72339756, #72339758 |
| maximal normal subgroup of | #72339712, #72339756, #72339758 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56752);