Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #510514 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #31680 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #24281 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #2929 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
614 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 614 conjugacy classes of subgroups — too many to draw. Only its 112 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:
(1,2,4,10)(3,8)(5,9,7,11)(6,12,17,27)(13,30)(14,23)(15,25)(16,22)(18,26,20,28)(19,29)(21,24)(31,32);(1,3,7,21,4,14,5,15)(2,8,11,24,10,23,9,25)(6,19,20,16,17,13,18,31)(12,29,28,22,27,30,26,32);(2,9)(3,13)(5,7)(6,18)(8,22)(10,11)(12,27)(14,19)(15,31)(16,21)(17,20)(23,32)(24,29)(25,30);(2,10)(3,14)(8,23)(9,11)(12,27)(13,19)(15,21)(16,31)(22,32)(24,25)(26,28)(29,30)Polycyclic presentation:
2^9|a^2=efh,b^2=ef,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=defh,e^a=ei,f^a=fi,c^b=cg,d^b=de,g^b=gh,h^b=hi,f^c=fi,g^c=ghi,h^c=hi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72237280, C2×#510514, #72320552, … |
| maximal normal subgroup of | #72237280, C2×#510514, #72320552, … |
| maximal quotient of | C2×#510514, #72250962, #72257153, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 417710);