Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #139712 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1898 |
| commutator | C25 | C22×C4 |
| Frattini, cosocle | C22×C22⋊C4 | C23 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C25 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×#123 ◁ G | |
| LECS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
704 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 704 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:
(1,2,5,11,24,15,8,26)(3,13,17,20,10,23,30,19)(4,16,14,27,32,18,29,21)(6,9,22,28,7,12,25,31);(1,3,14,28,6,16,13,11)(2,8,27,20,9,25,17,32)(4,15,5,21,19,12,22,30)(7,18,23,26,24,10,29,31);(2,9)(4,19)(5,22)(8,25)(12,15)(17,27)(20,32)(21,30)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,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=fg,g^b=gi,d^c=dh,f^c=fh,f^d=fi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72237447, #72238433, #72238435, … |
| maximal normal subgroup of | #72237447, #72238433, #72238435, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47138);