Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #139432 |
| 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 | C22×C4⋊C4 | C23 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×#123 ◁ G | |
| LECS | G ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
671 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 671 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:
(1,2,5,11,7,13,24,30)(3,14,16,31,10,29,21,18)(4,15,19,27,20,17,23,32)(6,9,22,28,25,12,8,26);(1,3,14,11,6,15,19,28)(2,8,27,20,9,24,16,31)(4,12,5,21,18,13,22,32)(7,10,29,30,25,17,23,26);(2,9)(4,18)(5,22)(8,24)(12,13)(16,27)(20,31)(21,32)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,d^2=ghi,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eghi,g^a=gi,h^a=hi,d^b=dh,e^b=eghi,f^b=fhi,g^b=gi,h^b=hi,d^c=dghi,f^c=fghi,f^d=fi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72238370, #72238434, #72238435, … |
| maximal normal subgroup of | #72238370, #72238434, #72238435, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 46672);