Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #130478 |
| 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 | #39338 | C2 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×D42 ◁ G | |
| LECS | G ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1 | |
1091 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 1091 conjugacy classes of subgroups — too many to draw. Only its 98 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,4,6,5,8,3,7)(9,10)(11,17)(12,20,21,15)(13,16)(14,19,23,24)(18,22);(2,6)(3,4)(7,8)(9,11,13,18)(10,14,16,23)(12,19,21,24)(15,17,20,22);(10,15)(11,18)(14,22)(16,20)(17,23)(19,24)Polycyclic presentation:
2^9|a^2=f,b^2=i,d^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,c^b=ci,d^b=dh,f^b=fgh,f^c=fg,f^d=fi,f^e=fi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72240240, #72240242, #72240250, … |
| maximal normal subgroup of | #72240240, #72240242, #72240250, … |
| maximal quotient of | #72240324 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 37942);