Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #486827 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×C22⋊C4 | C24 |
| prime radical | #52395 | C2 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
974 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 974 conjugacy classes of subgroups — too many to draw. Only its 118 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:
(1,2,5,8)(3,9,10,16)(4,12,14,7)(6,13,15,11)(17,18)(19,23)(20,22)(21,24)(25,32)(26,29)(27,31)(28,30);(1,3,5,10)(2,6,8,15)(4,13,14,11)(7,16,12,9)(18,22)(19,25)(23,31)(24,30)(26,27)(29,32);(3,10)(4,14)(6,15)(9,16)(17,19)(18,23)(20,27)(21,25)(22,29)(24,32)(26,28)(30,31);(19,25)(23,32)(26,27)(29,31)Polycyclic presentation:
2^9|a^2=g,b^2=g,e^2=g,b^a=be,c^a=cf,e^a=eg,h^a=hi,c^b=ci,e^b=eg,f^b=fhi,d^c=dgi,e^c=eh,f^e=fi41 conjugacy classes — view the full character table.
| maximal subgroup of | #72227761, #72227774 |
| maximal normal subgroup of | #72227761 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 394407);