Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #94554 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #7349 |
| commutator | C25 | C2×C8 |
| Frattini, cosocle | #1781 | C22 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C25 ▷ C24 ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C24 ◁ C25 ◁ #1781 ◁ G | |
| LECS | G ▷ #1781 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
226 conjugacy classes of subgroups; only the 24 normal subgroups are shown.
This group has 226 conjugacy classes of subgroups — too many to draw. Only its 24 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 — 2 generators on 32 points:
(1,2,5,9)(3,11,13,28)(4,15,16,31,17,8,19,25)(6,10,20,27)(7,21,22,29,23,30,26,12)(14,24,18,32);(1,3)(2,7,10,23)(4,12)(5,18,20,13)(6,14)(8,24,15,11)(9,26)(16,30,19,21)(17,29)(22,27)(25,28)(31,32)Polycyclic presentation:
2^9|a^2=d,b^2=g,d^2=f,b^a=bc,c^a=ce,e^a=eg,g^a=gh,h^a=hi,c^b=ch,d^b=de,e^b=ei,f^b=fh,d^c=dg,f^c=fi,e^d=eh,g^d=gi29 conjugacy classes — view the full character table.
| maximal subgroup of | #72283440, #72283621, #72283631, … |
| maximal normal subgroup of | #72283440, #72283621, #72283631, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1750);