Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #139718 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1898 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C23×D4 | C23 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×#121 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
827 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 827 conjugacy classes of subgroups — too many to draw. Only its 74 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)(3,14,18,32)(4,17,22,29)(5,24,26,16)(6,13)(7,11)(8,12)(9,21,28,23)(10,15,30,27)(19,25,20,31);(1,3,15,17)(2,4,16,31)(5,9,7,18)(6,20,27,28)(8,19,26,29)(10,25,11,23)(12,22,30,32)(13,21,24,14);(1,4,7,23)(2,9,11,17)(3,16,18,10)(5,25,15,31)(6,21,8,22)(12,28,13,29)(14,26,32,27)(19,30,20,24)Polycyclic presentation:
2^9|a^2=f,b^2=dfh,c^2=hi,d^2=i,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 | #72233107, #72233109, #72233113, … |
| maximal normal subgroup of | #72233107, #72233109, #72233113, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47047);