Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #153179 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13346 |
| commutator, Frattini, cosocle | #532 | C23 |
| prime radical | #33570 | C2 |
| LCS | G ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ G | |
| LECS | G ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
574 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 574 conjugacy classes of subgroups — too many to draw. Only its 50 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 — 3 generators on 32 points:
(1,2,7,12)(3,13,17,24)(4,11,6,9)(5,19,20,28)(8,16,25,15)(10,22,26,18)(14,23,21,27)(29,32,30,31);(3,14)(4,6)(5,20)(8,23)(9,11)(10,26)(13,29)(15,32)(16,31)(17,21)(24,30)(25,27);(1,3)(2,8)(4,16)(5,21)(6,15)(7,17)(9,24)(10,27)(11,13)(12,25)(14,20)(18,31)(19,30)(22,32)(23,26)(28,29)Polycyclic presentation:
2^9|a^2=i,d^2=hi,e^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dhi,e^a=ei,f^a=fghi,g^a=gi,c^b=cf,d^b=dhi,e^b=ei,f^b=fi,g^b=gh,d^c=dg,e^c=ei,f^c=fi,g^c=gi,h^c=hi,f^d=fh,g^f=gi35 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60377);