Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #94168 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #7213 |
| commutator | C2×C22⋊C4 | C42 |
| Frattini, cosocle | #2309 | C22 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C23×C4 ◁ #2309 ◁ G | |
| LECS | G ▷ #2309 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
358 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 358 conjugacy classes of subgroups — too many to draw. Only its 42 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)(3,11,13,23)(4,8,15,21)(5,17,18,9)(6,10)(7,12,19,25)(14,28)(16,30)(20,26)(22,29)(24,27)(31,32);(1,3,5,14,6,13,18,27)(2,7,9,20,10,19,17,31)(4,12,16,26,15,25,29,32)(8,11,22,24,21,23,30,28)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fg,e^2=i,b^a=bc,c^a=cf,e^a=efg,f^a=fh,g^a=ghi,d^b=dg,f^b=fi,g^b=gi,h^b=hi,d^c=dh,f^c=fi,g^c=gi,e^d=ei41 conjugacy classes — view the full character table.
| maximal subgroup of | #72213734, #72213776, #72213786, … |
| maximal normal subgroup of | #72213734, #72213776, #72213786, … |
| maximal quotient of | C2×#94169 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1365);