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×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1898 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C23×Q8 | C23 |
| prime radical | C24 | C2×C22⋊C4 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C4⋊Q8 ◁ G | |
| LECS | G ▷ C23×Q8 ▷ C23 ▷ C2 ▷ C1 | |
591 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 591 conjugacy classes of subgroups — too many to draw. Only its 76 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) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,6)(4,11,12,7)(5,8)(9,13,15,16)(10,14)(17,18,21,24,22,25,32,28)(19,26,23,31,29,20,30,27);(1,3,5,10)(2,6,8,14)(4,9,12,15)(7,13,11,16)(17,19,27,18,22,29,31,25)(20,28,32,30,26,24,21,23);(1,4,5,12)(2,7,8,11)(3,9,10,15)(6,13,14,16)Polycyclic presentation:
2^9|a^2=f,b^2=dfh,c^2=h,d^2=i,f^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 | #72271333, #72271343, #72271347 |
| maximal normal subgroup of | #72271333, #72271343, #72271347 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47048);