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 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ G | |
| LECS | G ▷ C4⋊Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
1041 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 1041 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:
(2,7)(3,13)(4,18)(5,20)(8,26)(9,10)(11,25)(12,21)(14,31)(15,16)(17,22)(24,27)(28,30)(29,32);(1,2,6,12)(3,8,17,28)(4,11,18,9)(5,21,20,7)(10,19,25,23)(13,24,22,32)(14,30,16,26)(15,29,31,27);(1,3)(2,8)(4,14)(5,22)(6,17)(7,24)(9,26)(10,29)(11,30)(12,28)(13,20)(15,23)(16,18)(19,31)(21,32)(25,27);(1,4,6,18)(2,9,12,11)(3,14,17,16)(5,19,20,23)(7,25,21,10)(8,26,28,30)(13,15,22,31)(24,27,32,29)Polycyclic presentation:
2^9|b^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,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.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420101);