Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #133077 |
| 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 | C23×D4 | D4 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×#123 ◁ G | |
| LECS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
900 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 900 conjugacy classes of subgroups — too many to draw. Only its 86 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 — 3 generators on 32 points:
(1,2)(3,10,12,7)(4,6)(5,9)(8,13)(11,15,16,14)(17,18)(19,25)(20,28,29,23)(21,24)(22,26,30,32)(27,31);(2,6)(4,13)(7,14)(8,9)(10,15)(11,16)(17,19,21,27)(18,22,24,30)(20,26,29,32)(23,25,28,31);(1,3)(2,7)(4,11)(5,12)(6,14)(8,15)(9,10)(13,16)Polycyclic presentation:
2^9|a^2=f,b^2=i,d^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,d^b=dh,f^b=fgh,f^c=fh,f^d=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72292874, C2×#133078 |
| maximal normal subgroup of | #72292874, C2×#133078 |
| maximal quotient of | C2×#133078, #72293068 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 40274);