Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #512827 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D16 |
| commutator, Frattini, cosocle | C2×C16 | C24 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
496 conjugacy classes of subgroups; only the 140 normal subgroups are shown.
This group has 496 conjugacy classes of subgroups — too many to draw. Only its 140 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6)(7,11)(8,20)(9,23)(10,12)(13,16)(14,17)(15,21)(18,30)(19,24)(22,36)(25,32)(26,31)(27,33)(28,34)(29,35);(1,2,3,4);(6,11)(7,16)(8,17)(9,21)(12,24)(13,14)(15,30)(18,35)(19,33)(20,25)(22,34)(23,26)(27,28)(29,36)(31,32);(6,12)(11,24)(13,27)(14,28)(15,23)(20,36)(25,29)(26,30)Polycyclic presentation:
2^9|b^2=e,f^2=gh,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,f^a=fg,g^a=gh,h^a=hi,f^c=fg,g^c=gh,h^c=hi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72225449, #72225508, #72225513, … |
| maximal normal subgroup of | #72225449, #72225508, #72225513, … |
| maximal quotient of | #72225429, #72225449 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420110);