Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11992 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1555 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C4×C8 | C24 |
| socle | C2 | C2×#1555 |
| prime radical | #60639 | C2 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
1015 conjugacy classes of subgroups; only the 208 normal subgroups are shown.
This group has 1015 conjugacy classes of subgroups — too many to draw. Only its 208 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 — 4 generators on 32 points:
(1,2,5,10,4,9,17,25,6,8,20,24,18,11,28,27)(3,12,14,29,15,21,31,32,16,22,30,26,13,7,19,23);(1,3)(2,7)(4,13)(5,19)(6,16)(8,21)(9,22)(10,26)(11,12)(14,28)(15,18)(17,30)(20,31)(23,27)(24,29)(25,32);(2,8)(5,20)(7,21)(9,11)(12,22)(14,30)(17,28)(19,31);(1,4,6,18)(2,9,8,11)(3,13,16,15)(5,17,20,28)(7,22,21,12)(10,25,24,27)(14,19,30,31)(23,26,32,29)Polycyclic presentation:
2^9|a^2=f,d^2=i,e^2=h,f^2=gh,g^2=i,b^a=be,c^a=ch,e^a=eg,g^a=gi,h^a=hi,e^b=eh,f^b=fghi,g^b=gi,f^c=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72214114, #72214208, #72214702, … |
| maximal normal subgroup of | #72214114, #72214208, #72214702, … |
| maximal quotient of | #72224797, #72226777, C4×#11992 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 390500);