Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11663 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#102 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C23×D8 | C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
1074 conjugacy classes of subgroups; only the 158 normal subgroups are shown.
This group has 1074 conjugacy classes of subgroups — too many to draw. Only its 158 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) = 128 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2)(3,4)(5,6,7,9,8,10,11,12)(13,14,16,20,17,22,27,21)(15,23,25,19,26,18,24,28);(14,18)(15,24)(16,27)(19,20)(21,28)(22,23)(25,26);(2,4)Polycyclic presentation:
2^9|a^2=f,d^2=g,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,d^b=dgi,f^b=fi,g^b=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72225044, #72225051, #72225063, … |
| maximal normal subgroup of | #72225044, #72225051, #72225063, … |
| maximal quotient of | #72225033, #72225036, #72225039, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36699);