Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11677 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#102 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C22×D8 | C2×C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
414 conjugacy classes of subgroups; only the 130 normal subgroups are shown.
This group has 414 conjugacy classes of subgroups — too many to draw. Only its 130 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2,4,3)(5,6,8,11)(7,9,12,10)(14,20)(15,16)(17,18)(21,23)(22,25)(24,28)(26,27);(6,9,11,10)(13,14,16,22,18,24,27,28,19,25,26,20,17,23,15,21);(1,3,4,2)(13,15,17,26,19,27,18,16)(14,21,23,20,25,28,24,22)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=eg,d^2=gi,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ch,d^a=dg,f^a=fg,g^a=gi,h^a=hi,d^b=di104 conjugacy classes — view the full character table.
| maximal subgroup of | #72318459, #72318470, #72318476, … |
| maximal normal subgroup of | #72318459, #72318470, #72318476, … |
| maximal quotient of | #72322438, #72322472 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35799);