Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #485328 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×C4⋊C4 | C24 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ G | |
| LECS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1087 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 1087 conjugacy classes of subgroups — too many to draw. Only its 120 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 — 4 generators on 32 points:
(1,2)(3,15)(4,8)(5,23)(6,10)(7,14)(9,18)(11,20)(12,17)(13,26)(16,30)(19,32)(21,29)(22,28)(24,27)(25,31);(2,8)(3,16)(4,20)(5,21)(9,13)(10,11)(12,30)(14,28)(15,23)(17,25)(18,22)(19,24)(27,29)(31,32);(1,3)(2,9,14,11)(4,21)(5,18)(6,19)(7,16)(8,10,28,13)(12,15,23,30)(17,20)(22,25)(24,26)(27,29,32,31);(2,10)(8,11)(9,28)(12,29)(13,14)(15,32)(23,31)(27,30)Polycyclic presentation:
2^9|c^2=gh,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fi,h^a=hi,c^b=ci,e^b=eg,f^b=fhi,e^c=eh,f^e=fi41 conjugacy classes — view the full character table.
| maximal subgroup of | #72251767, #72251999, #72252003, … |
| maximal normal subgroup of | #72251767, #72251999, #72252003, … |
| maximal quotient of | #72278647, #72279163 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 392606);