Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #489347 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #18524 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C4×M16 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
907 conjugacy classes of subgroups; only the 124 normal subgroups are shown.
This group has 907 conjugacy classes of subgroups — too many to draw. Only its 124 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:
(3,11)(4,15)(5,19)(7,21)(8,16)(9,20)(12,17)(13,28)(14,29)(18,25)(22,26)(23,31)(24,32)(27,30);(1,2,6,10)(3,7,13,23)(4,16,18,30)(5,20,19,9)(8,25,27,15)(11,26,28,32)(12,24,14,22)(17,21,29,31);(1,3)(2,7)(4,17)(5,12)(6,13)(8,26)(9,22)(10,23)(11,15)(14,19)(16,21)(18,29)(20,24)(25,28)(27,32)(30,31);(4,18)(8,27)(11,28)(15,25)(16,30)(17,29)(21,31)(26,32)Polycyclic presentation:
2^9|b^2=i,e^2=g,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fh,h^a=hi,e^b=eg,f^b=fh,h^b=hi,e^c=eh,f^c=fh,g^c=gi,h^c=hi,f^d=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72214208, #72214826, #72215848, … |
| maximal normal subgroup of | #72214208, #72214826, #72215848, … |
| maximal quotient of | #72217755, #72217759, #72217768, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 396544);