Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12314 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1619 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| socle | C2 | C2×#1619 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
968 conjugacy classes of subgroups; only the 152 normal subgroups are shown.
This group has 968 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:
(1,2)(3,12)(4,9)(5,18)(6,23)(7,11)(8,15)(10,17)(13,27)(14,31)(16,30)(19,22)(20,26)(21,28)(24,29)(25,32);(1,2)(3,8)(4,9)(5,19)(6,23)(7,11)(10,21)(12,29)(13,24)(14,26)(15,31)(16,25)(17,28)(18,22)(20,30)(27,32);(1,3)(2,8)(4,13)(5,20)(6,14)(7,16)(9,24)(10,27)(11,25)(12,18)(15,17)(19,30)(21,32)(22,29)(23,26)(28,31);(1,4,7,6)(2,9,11,23)(3,13,16,14)(5,17,22,21)(8,24,25,26)(10,19,28,18)(12,27,30,31)(15,29,32,20)Polycyclic presentation:
2^9|d^2=i,e^2=g,f^2=h,h^2=i,b^a=be,c^a=cf,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 | #72214466, #72214690, #72214826, … |
| maximal normal subgroup of | #72214466, #72214690, #72214826, … |
| maximal quotient of | #72224798, C4×#12314 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 396072);