Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #128057 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#342 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C22×C4⋊C4 | C23 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
816 conjugacy classes of subgroups; only the 130 normal subgroups are shown.
This group has 816 conjugacy classes of subgroups — too many to draw. Only its 130 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:
(2,8,10,25)(3,14)(4,18)(5,22)(9,28,11,26)(12,27,13,19)(15,24,29,30)(17,20)(21,32)(23,31);(1,2,5,11,6,12,3,15)(4,19,20,30,18,25,31,28)(7,13,22,29,16,10,14,9)(8,23,26,32,27,17,24,21);(1,3,16,22)(2,9,10,11)(4,17,32,20)(5,7,14,6)(8,24,25,30)(12,29,13,15)(18,23,21,31)(19,28,27,26)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=gi,d^2=i,f^2=g,b^a=bd,c^a=ci,d^a=dg,f^a=fhi,c^b=ch,d^b=dh,e^b=egi,h^b=hi,f^c=fi,f^d=fi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72231339, #72231376, #72231381, … |
| maximal normal subgroup of | #72231339, #72231376, #72231381, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35396);