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 |
| 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 | |
1036 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 1036 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)(3,12)(4,17)(5,22)(6,9)(7,13)(8,14)(10,21)(11,25)(15,29)(16,30)(18,26)(19,28)(20,31)(23,27)(24,32);(1,3)(2,7)(4,18)(5,15)(6,16)(8,22)(9,26)(10,30)(11,28)(12,17)(13,21)(14,20)(19,24)(23,25)(27,31)(29,32);(1,3)(2,8)(4,19)(5,23)(6,16)(7,22)(9,28)(10,30)(11,26)(12,17)(13,20)(14,21)(15,25)(18,24)(27,32)(29,31);(2,9)(7,26)(8,28)(10,17)(11,22)(12,30)(27,29)(31,32)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,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 | #72283007, #72283015, #72283075, … |
| maximal normal subgroup of | #72283007, #72283015, #72283075, … |
| maximal quotient of | #72340679, #72340681 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420090);