Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12366 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1629 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C2 | #8443 |
| prime radical | #61105 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
790 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 790 conjugacy classes of subgroups — too many to draw. Only its 112 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:
(1,2,6,12,10,13,23,25,8,14,5,21,7,4,20,11)(3,15,17,31,26,9,24,28,19,16,32,27,18,29,22,30);(1,3)(2,9)(4,16)(5,22)(6,24)(7,26)(8,19)(10,18)(11,30)(12,27)(13,15)(14,29)(17,23)(20,32)(21,31)(25,28);(1,4)(2,10)(3,16)(5,11)(6,25)(7,14)(8,13)(9,18)(12,20)(15,19)(17,31)(21,23)(22,30)(24,28)(26,29)(27,32)Polycyclic presentation:
2^9|a^2=f,d^2=h,e^2=gi,f^2=ghi,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,g^a=gi,h^a=hi,d^b=dh,e^b=eg,f^b=fghi,g^b=gi,d^c=dg,e^c=eg,f^c=fi,g^c=gi,h^c=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72214124, #72214153 |
| maximal normal subgroup of | #72214124, #72214153 |
| maximal quotient of | #72216470, #72225233 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 39035);