Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #139718 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1898 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C22×C4⋊C4 | C23 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×#123 ◁ G | |
| LECS | G ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
697 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 697 conjugacy classes of subgroups — too many to draw. Only its 72 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,6,9,5,11,23,31)(3,4,15,21,17,20,25,32)(7,12,26,14,24,10,30,19)(8,27,18,22,29,13,16,28);(1,3,13,11,5,17,27,2)(4,19,6,25,20,14,23,15)(7,18,21,10,24,16,32,12)(8,28,9,30,29,22,31,26);(4,20)(6,23)(8,29)(9,31)(14,19)(15,25)(22,28)(26,30)Polycyclic presentation:
2^9|a^2=f,b^2=dfi,d^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,d^b=dgi,e^b=eh,f^b=fgi,g^b=gi,d^c=dh,f^c=fh,f^d=fi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72237579, #72237582, #72237584, … |
| maximal normal subgroup of | #72237579, #72237582, #72237584, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47112);