Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12152 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | B(4,𝔽2) |
| commutator | C2×C4⋊C4 | C22×C4 |
| Frattini, cosocle | C2×#117 | C23 |
| socle | C22 | #1790 |
| prime radical | #34976 | C2 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1 | |
880 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 880 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) = 32 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,4,6,5,8,3,7)(9,10)(11,19)(12,16,22,15)(13,17,20,24)(14,18)(21,23);(2,6)(3,4)(7,8)(10,15)(12,22)(13,20)(16,18)(17,19)(23,24);(9,11,14,21)(10,15)(12,20,22,13)(16,18)(17,19)(23,24)Polycyclic presentation:
2^9|a^2=f,c^2=g,d^2=h,e^2=i,f^2=hi,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,d^b=dh,e^b=eg,f^b=fgh,d^c=dgi,f^c=fgi,e^d=ei,f^d=fi,f^e=fi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72338383, #72338389, #72338392, … |
| maximal normal subgroup of | #72338383, #72338389, #72338392, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 38576);