Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509858 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #31064 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #2929 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
868 conjugacy classes of subgroups; only the 128 normal subgroups are shown.
This group has 868 conjugacy classes of subgroups — too many to draw. Only its 128 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:
(1,2,7,22,6,12,4,11,5,9,18,30,19,27,20,10)(3,13,14,8,16,29,23,26,17,31,32,25,21,28,15,24);(1,3)(2,8)(4,14)(5,17)(6,21)(7,23)(9,25)(10,28)(11,29)(12,24)(13,30)(15,18)(16,19)(20,32)(22,31)(26,27);(2,9)(10,11)(12,27)(22,30);(2,10)(3,14)(4,18)(6,19)(7,20)(8,24)(9,11)(12,30)(15,16)(17,32)(21,23)(22,27)(25,26)(28,29)Polycyclic presentation:
2^9|a^2=efh,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=defh,g^a=gi,c^b=cg,d^b=de,e^b=eh,f^b=fi,h^b=hi,e^d=eh,h^d=hi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72237280, C2×#509858, #72320552, … |
| maximal normal subgroup of | #72237280, C2×#509858, #72320552, … |
| maximal quotient of | C2×#509858, #72250957, #72257159, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 417054);