Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #152103 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #12696 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C2×#127 | C23 |
| prime radical | #60798 | C2 |
| LCS | G ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C23×C4 ◁ C22×#127 ◁ G | |
| LECS | G ▷ C2×#127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
872 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 872 conjugacy classes of subgroups — too many to draw. Only its 76 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2)(3,10,12,6)(4,8,14,7)(5,9)(11,16,13,15);(1,3)(2,6)(4,13)(5,11)(7,16)(8,10)(9,15)(12,14);(3,11)(12,13)Polycyclic presentation:
2^9|a^2=f,d^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gh,h^a=hi,c^b=ch,d^b=dhi,e^b=ei,f^b=fgh,g^b=gi,f^d=fhi,g^f=gi38 conjugacy classes — view the full character table.
| maximal subgroup of | #72213428, #72216600, #72216603, … |
| maximal normal subgroup of | #72213428, #72216600, #72216603, … |
| maximal quotient of | C2×#152103, #72279361 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59299);