Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #133093 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1781 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C22×C22⋊C4 | C23 |
| prime radical | C2×#392 | C22 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×#126 ◁ G | |
| LECS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
974 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 974 conjugacy classes of subgroups — too many to draw. Only its 82 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 — 3 generators on 32 points:
(1,2)(3,13,15,8)(4,18,20,7)(5,11)(6,12)(9,19,23,32)(10,21)(14,24,29,22)(16,26,17,25)(27,31,28,30);(2,7)(4,19)(8,22)(9,11)(10,23)(12,18)(13,27)(14,29)(20,32)(24,25)(26,28)(30,31);(1,3,6,17)(2,8,12,26)(4,14,20,30)(5,16,21,15)(7,22,18,28)(9,24,23,27)(10,13,11,25)(19,29,32,31)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,g^a=gi,d^b=dh,f^b=fgh,f^c=fhi,f^d=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72292874, C2×#133095 |
| maximal normal subgroup of | #72292874, C2×#133095 |
| maximal quotient of | C2×#133095, #72293065 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 40291);