Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #512902 |
| 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 |
| LCS | G ▷ C4⋊Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C22×C4 ◁ C2×C4⋊Q8 ◁ G | |
| LECS | G ▷ C4⋊Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
1016 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 1016 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)(4,15)(5,19)(6,24)(7,12)(8,9)(10,23)(11,22)(14,26)(16,31)(17,32)(18,20)(21,30)(25,28)(27,29);(1,3)(2,4)(5,20)(6,25)(7,16)(8,14)(9,18)(10,26)(11,21)(12,17)(13,19)(15,23)(22,32)(24,31)(27,30)(28,29);(1,4)(2,3)(5,21)(6,26)(7,28)(8,18)(9,14)(10,25)(11,20)(12,30)(13,22)(15,24)(16,29)(17,27)(19,32)(23,31);(2,9)(3,14)(10,19)(11,24)(12,29)(15,20)(16,30)(25,32)Polycyclic presentation:
2^9|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,d^b=di,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 | #72306631, #72306632, #72306637, … |
| maximal normal subgroup of | #72306631, #72306632, #72306637, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420098);