Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #512894 |
| 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 | C4⋊Q8 | C24 |
| prime radical | #61105 | C2 |
| LCS | G ▷ C4⋊Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ G | |
| LECS | G ▷ C4⋊Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
1018 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 1018 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:
(1,2,9,13)(3,14,19,21)(4,17,22,16)(5,23,8,10)(6,25,24,11)(7,27,29,12)(15,26,18,32)(20,31,30,28);(1,3)(2,4)(5,15)(6,17)(7,28)(8,18)(9,19)(10,20)(11,21)(12,32)(13,22)(14,25)(16,24)(23,30)(26,27)(29,31);(1,4)(2,3)(5,20)(6,26)(7,14)(8,30)(9,22)(10,15)(11,31)(12,16)(13,19)(17,27)(18,23)(21,29)(24,32)(25,28);(1,5,9,8)(2,10,13,23)(3,15,19,18)(4,20,22,30)(6,7,24,29)(11,27,25,12)(14,32,21,26)(16,31,17,28)Polycyclic presentation:
2^9|a^2=i,d^2=i,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=ei,f^a=fi,g^a=gh,e^b=ei,f^b=fghi,h^b=hi,e^c=eg,f^c=fi,g^c=gi,h^c=hi,f^e=fhi,g^e=gi,g^f=gi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72283143, #72283226, #72283253, … |
| maximal normal subgroup of | #72283143, #72283226, #72283253, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420093);