Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #485056 |
| 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 |
| prime radical | #23948 | C2 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ G | |
| LECS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1029 conjugacy classes of subgroups; only the 124 normal subgroups are shown.
This group has 1029 conjugacy classes of subgroups — too many to draw. Only its 124 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)(3,12)(4,17)(5,9)(6,10)(7,14)(8,15)(11,19)(13,21)(16,29)(18,24)(20,30)(22,26)(23,28)(25,31)(27,32);(1,3,6,16)(2,8,10,24)(4,18,19,15)(5,21,23,7)(9,27,28,30)(11,29,17,12)(13,25,14,26)(20,31,32,22);(2,9)(3,5)(4,19)(8,11)(10,28)(12,14)(13,29)(15,31)(16,23)(17,24)(18,22)(20,32)(25,30)(26,27);(2,10)(8,24)(9,28)(11,17)(12,29)(13,14)(25,26)(27,30)Polycyclic presentation:
2^9|b^2=g,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^c=fi,f^e=fi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72240210, #72240300, #72251767, … |
| maximal normal subgroup of | #72240210, #72240300, #72251767, … |
| maximal quotient of | #72240578 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 392290);