Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #123275 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1619 |
| commutator, Frattini, cosocle | C82 | C23 |
| LCS | G ▷ C82 ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C82 ◁ G | |
| LECS | G ▷ C82 ▷ C42 ▷ C22 ▷ C1 | |
341 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 341 conjugacy classes of subgroups — too many to draw. Only its 54 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) = 128 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,6)(4,13)(5,9)(7,10)(8,11)(12,15)(14,16)(17,18)(19,22)(20,29)(21,25)(23,26)(24,27)(28,31)(30,32);(2,6)(3,10)(4,11)(7,8)(9,15)(12,16)(13,14)(18,22)(19,26)(20,27)(23,24)(25,31)(28,32)(29,30);(18,22)(19,26)(20,27)(23,24)(25,31)(28,32)(29,30)Polycyclic presentation:
2^9|d^2=fh,e^2=gi,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=df,e^a=eg,f^a=fh,g^a=gi,d^b=df,e^b=eg,f^b=fh,g^b=gi,d^c=dg,e^c=eg,f^c=fi,g^c=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72214276, C2×#123275, #72225210, … |
| maximal normal subgroup of | #72214276, C2×#123275, #72225210, … |
| maximal quotient of | #72214187, #72214214, C2×#123275, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30471);