Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #486803 |
| 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 ◁ C24 ◁ C22×C22⋊C4 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1155 conjugacy classes of subgroups; only the 116 normal subgroups are shown.
This group has 1155 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) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,8)(4,7)(5,10)(6,11)(9,15)(12,23)(13,21)(14,18)(16,19)(17,25)(20,27)(22,28)(24,30)(26,32)(29,31);(2,7)(3,12)(4,15)(8,20)(9,10)(11,19)(13,26)(14,29)(16,30)(18,23)(21,31)(22,32)(24,25)(27,28);(3,13)(4,16)(8,21)(12,26)(14,29)(15,30)(18,22)(20,31)(23,32)(27,28);(1,3)(2,8)(4,14)(5,13)(6,12)(7,18)(9,22)(10,21)(11,23)(15,28)(16,27)(17,26)(19,20)(24,31)(25,32)(29,30)Polycyclic presentation:
2^9|e^2=g,b^a=be,c^a=cf,e^a=eg,h^a=hi,d^b=di,e^b=eg,f^b=fhi,d^c=dg,e^c=eh,f^e=fi41 conjugacy classes — view the full character table.
| maximal subgroup of | #72231830, #72292039 |
| maximal normal subgroup of | #72231830, #72292039 |
| maximal quotient of | #72292049 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 393999);