Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #512880 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#2196 |
| commutator, Frattini, cosocle | #124 | C24 |
| LCS | G ▷ #124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ G | |
| LECS | G ▷ #124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
1049 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 1049 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)(3,13)(4,9)(5,11)(6,12)(7,15)(8,16)(10,23)(14,28)(17,25)(18,31)(19,30)(20,24)(21,27)(22,29)(26,32);(1,3)(2,7)(4,17)(5,21)(6,18)(8,24)(9,26)(10,31)(11,13)(12,27)(14,20)(15,23)(16,19)(22,32)(25,30)(28,29);(1,3)(2,8)(4,14)(5,16)(6,18)(7,24)(9,27)(10,31)(11,13)(12,26)(15,22)(17,20)(19,21)(23,32)(25,29)(28,30);(1,4,6,20)(2,9,12,24)(3,14,18,17)(5,19,23,22)(7,8,27,26)(10,29,11,30)(13,28,31,25)(15,16,21,32)Polycyclic presentation:
2^9|d^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,e^a=ei,g^a=gh,d^b=di,e^b=ei,f^b=fghi,h^b=hi,e^c=eg,g^c=gi,h^c=hi,f^e=fhi,g^e=gi,g^f=gi41 conjugacy classes — view the full character table.
| maximal subgroup of | #72340538 |
| maximal normal subgroup of | #72340538 |
| maximal quotient of | #72340621, #72340624, #72340683, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420078);