Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7221 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #340 |
| commutator | C2×C8 | C4×C8 |
| Frattini, cosocle | C2×C82 | C22 |
| socle | C22 | #1279 |
| prime radical | C23 | Q8⋊C8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C82 ▷ C42 ▷ C22 ▷ C1 | |
173 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 173 conjugacy classes of subgroups — too many to draw. Only its 74 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) = 256 |
A minimal faithful permutation representation — 2 generators on 32 points:
(2,6)(3,10)(4,11)(7,9)(8,15)(12,16)(13,14)(17,18,20,23,21,26,31,32,22,27,19,28,29,25,30,24);(1,2,3,7,11,13,16,14,5,9,12,6,4,8,10,15)(18,23,25,24,27,28,26,32)(19,20)(30,31)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fhi,d^2=g,e^2=fgh,f^2=h,g^2=i,b^a=bc,c^a=cf,e^a=efi,f^a=fh,c^b=ch,d^b=di80 conjugacy classes — view the full character table.
| maximal subgroup of | #72216552, #72216593, #72216594, … |
| maximal normal subgroup of | #72216552, #72216593, #72216594, … |
| maximal quotient of | #72216538 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 140);