Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #108535 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C4×C4⋊C4 |
| commutator | C42 | C2×C42 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1767 |
| prime radical | C23 | #432 |
| LCS | G ▷ C42 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#322 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
244 conjugacy classes of subgroups; only the 114 normal subgroups are shown.
This group has 244 conjugacy classes of subgroups — too many to draw. Only its 114 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) = 128 |
A minimal faithful permutation representation — 3 generators on 32 points:
(2,6,8,11)(3,9)(7,14,15,13)(10,16)(18,22)(19,26)(23,25)(27,30);(1,2)(3,7)(4,11)(5,8)(6,12)(9,15)(10,14)(13,16)(17,18,19,24,21,25,27,32,20,22,26,29,28,23,30,31);(1,3,4,10,5,9,12,16)(2,7,11,14,8,15,6,13)(18,23,22,25)(19,26)(24,32,29,31)(27,30)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=de,d^2=gi,f^2=h,h^2=i,b^a=bd,c^a=cgi,f^a=fi,c^b=ch,d^b=dg,e^b=egi,f^c=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72214602, #72214681, #72214696, … |
| maximal normal subgroup of | #72214602, #72214681, #72214696, … |
| maximal quotient of | #72214538, #72216252 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36280);