Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #153171 |
| 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 |
| 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:
(3,12)(4,17)(5,18)(7,21)(8,10)(9,25)(13,29)(14,16)(19,20)(22,23)(24,26)(28,31);(1,2)(3,13)(4,8)(5,9)(6,11)(7,15)(10,17)(12,27)(14,30)(16,29)(18,24)(19,26)(20,25)(21,28)(22,31)(23,32);(1,3)(2,7)(4,14)(5,15)(6,16)(8,22)(9,13)(10,21)(11,23)(12,17)(18,31)(19,32)(20,28)(24,30)(25,27)(26,29)Polycyclic presentation:
2^9|d^2=hi,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dhi,e^a=ei,f^a=fgh,g^a=gi,c^b=cf,d^b=dhi,g^b=gh,d^c=dg,e^c=ei,g^c=gi,h^c=hi,f^d=fh,g^f=gi32 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60367);