Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #488877 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#1619 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
989 conjugacy classes of subgroups; only the 152 normal subgroups are shown.
This group has 989 conjugacy classes of subgroups — too many to draw. Only its 152 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:
(3,12)(4,6)(5,16)(8,22)(9,21)(10,26)(13,18)(14,30)(15,31)(17,28)(19,20)(23,29)(24,32)(25,27);(1,2)(3,8)(4,9)(5,17)(6,21)(7,11)(10,19)(12,29)(13,23)(14,25)(15,24)(16,28)(18,32)(20,26)(22,30)(27,31);(1,3)(2,8)(4,13)(5,18)(6,14)(7,15)(9,23)(10,27)(11,24)(12,16)(17,32)(19,31)(20,30)(21,25)(22,26)(28,29);(1,4,7,6)(2,9,11,21)(3,13,15,14)(5,16,20,19)(8,23,24,25)(10,17,28,26)(12,30,31,18)(22,27,32,29)Polycyclic presentation:
2^9|d^2=i,e^2=g,f^2=h,h^2=i,b^a=be,c^a=cf,d^a=di,e^a=eg,f^a=fhi,h^a=hi,e^b=eg,f^b=fh,h^b=hi,e^c=eh,f^c=fhi,g^c=gi,h^c=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72214208, #72214645, #72214714, … |
| maximal normal subgroup of | #72214208, #72214645, #72214714, … |
| maximal quotient of | #72224805, C2×#488878, #72226786 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 396074);