Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #139702 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1889 |
| commutator | C22×D4 | C22×C4 |
| Frattini, cosocle | C23×D4 | C23 |
| prime radical | #3484 | C22 |
| LCS | G ▷ C22×D4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
767 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 767 conjugacy classes of subgroups — too many to draw. Only its 68 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:
(1,2,4,9)(3,8,13,22)(5,10,15,7)(6,11,17,25)(12,24,16,31)(14,23,30,19)(18,26,29,20)(21,27,32,28);(2,7,10,9)(3,12)(4,15)(8,21,23,31)(11,20,26,25)(13,28)(14,27)(16,30)(17,29)(19,32,22,24);(1,3,5,14)(2,8,10,23)(4,16,15,28)(6,12,18,27)(7,19,9,22)(11,24,26,32)(13,29,30,17)(20,21,25,31)Polycyclic presentation:
2^9|a^2=f,b^2=df,c^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,c^b=ci,d^b=dg,e^b=ehi,f^b=fg,d^c=dh,f^c=fh,e^d=ei,f^e=fi41 conjugacy classes — view the full character table.
| maximal subgroup of | #72233093, #72233111, #72233113, … |
| maximal normal subgroup of | #72233093, #72233111, #72233113, … |
| maximal quotient of | #72237059, #72251750, #72251806, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 46960);