Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #486827 |
| 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 | |
1142 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 1142 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) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,8)(4,7)(5,10)(6,11)(9,16)(12,26)(13,23)(14,20)(15,25)(17,22)(18,21)(19,28)(24,30)(27,31)(29,32);(2,7)(3,12)(4,16)(8,22)(9,10)(11,21)(13,15)(14,29)(17,30)(18,31)(20,26)(23,24)(25,32)(27,28);(1,3,6,12)(2,8,11,26)(4,17,18,14)(5,15,19,13)(7,20,21,22)(9,24,27,32)(10,25,28,23)(16,29,31,30);(3,13)(8,23)(12,15)(14,29)(17,30)(20,32)(22,24)(25,26)Polycyclic presentation:
2^9|c^2=i,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 | #72292041, #72307974, #72307990, … |
| maximal normal subgroup of | #72292041, #72307974, #72307990, … |
| maximal quotient of | #72292044, #72292050 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 394025);