Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #130474 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1781 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C2×C4×D4 | C23 |
| prime radical | #36212 | C2 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×D42 ◁ G | |
| LECS | G ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1 | |
1597 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 1597 conjugacy classes of subgroups — too many to draw. Only its 94 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) = 64 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2,6,7,5,3,4,8)(9,10)(11,15,16,13)(12,14);(1,3,5,2)(4,8,6,7)(10,13)(14,15);(1,4,5,6)(2,7,3,8)(10,13)(14,15)Polycyclic presentation:
2^9|a^2=f,b^2=h,c^2=h,d^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,d^b=dh,f^b=fgh,f^c=fg,f^d=fi,f^e=fi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72240201, #72240202, #72240212, … |
| maximal normal subgroup of | #72240201, #72240202, #72240212, … |
| maximal quotient of | #72240324, #72282765 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 38172);