Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #139718 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C42 |
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 | C2×#392 | C4 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×#126 ◁ G | |
| LECS | G ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
704 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 704 conjugacy classes of subgroups — too many to draw. Only its 70 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:
(1,2,7,9,27,11,4,20)(3,5,17,14,16,32,10,31)(6,13,23,30,8,12,22,24)(15,29,18,26,28,21,19,25);(1,3,14,11,27,16,31,2)(4,15,25,20,7,28,26,9)(5,24,23,10,32,30,22,17)(6,19,29,12,8,18,21,13);(1,4,6,22)(2,9,13,30)(3,15,19,17)(5,14,25,29)(7,8,23,27)(10,16,28,18)(11,20,12,24)(21,32,31,26)Polycyclic presentation:
2^9|a^2=f,b^2=df,c^2=i,d^2=i,f^2=hi,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 | #72235151 |
| maximal normal subgroup of | #72235151 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47115);