Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #133904 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1782 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C23×D4 | C23 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ #2309 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
865 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 865 conjugacy classes of subgroups — too many to draw. Only its 82 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)(3,7,8,5)(4,6)(9,10)(11,15,19,24)(12,20,22,16)(13,17)(14,18)(21,23);(9,11)(10,14)(12,21)(13,19)(15,20)(16,24)(17,23)(18,22);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15,17,24)(12,21)(13,19)(14,20,23,16)(18,22)Polycyclic presentation:
2^9|a^2=f,c^2=g,d^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,d^b=di,e^b=ei,f^b=fgi,d^c=di,e^c=ei,f^c=fh,f^d=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72261073, #72261355 |
| maximal normal subgroup of | #72261073, #72261355 |
| maximal quotient of | #72261484 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 41154);