Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #133877 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1781 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C23×Q8 | C23 |
| prime radical | C2×#2657 | C2 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×#124 ◁ G | |
| LECS | G ▷ C23×Q8 ▷ C23 ▷ C2 ▷ C1 | |
1466 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 1466 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) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,7,8,5)(4,6)(9,10,13,16,14,18,11,19)(12,15,20,24,21,17,22,23);(10,15,18,17)(11,13)(16,24,19,23)(20,22);(1,3,4,8)(2,5,6,7)(9,11)(10,16)(12,20)(13,14)(15,23)(17,24)(18,19)(21,22)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=hi,d^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,c^b=ci,f^b=fgi,f^c=fh,f^d=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72235017, #72235020, #72266388 |
| maximal normal subgroup of | #72235017, #72235020, #72266388 |
| maximal quotient of | #72266397 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 41214);