Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #486795 |
| 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×C22⋊C4 | C24 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
954 conjugacy classes of subgroups; only the 116 normal subgroups are shown.
This group has 954 conjugacy classes of subgroups — too many to draw. Only its 116 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) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,6)(4,11)(7,13)(8,12)(10,15)(14,16)(17,18)(19,23)(20,22)(21,24)(25,30)(26,28)(27,29)(31,32);(1,2,5,8)(3,7,9,14)(4,12,11,6)(10,16,15,13)(18,22)(23,28)(24,29)(30,32);(1,3)(2,7)(4,10)(5,9)(6,13)(8,14)(11,15)(12,16)(19,25)(20,27)(23,30)(28,32);(3,9)(7,14)(10,15)(13,16)(17,19,21,25)(18,23,24,30)(20,26,27,31)(22,28,29,32)Polycyclic presentation:
2^9|b^2=g,d^2=i,e^2=g,b^a=be,c^a=cf,e^a=eg,h^a=hi,e^b=eg,f^b=fhi,d^c=dgi,e^c=eh,f^e=fi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72307906, #72308007, #72308013, … |
| maximal normal subgroup of | #72307906, #72308007, #72308013, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 394251);