Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #153187 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13348 |
| commutator, Frattini, cosocle | #532 | C23 |
| prime radical | #3018 | C22 |
| 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 | |
484 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 484 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)(3,5)(4,14,18,26)(6,9)(7,23)(8,16,12,21)(10,22)(11,19)(13,25)(15,24,27,17)(20,28)(29,30,32,31);(1,3,5,2)(4,15,18,27)(6,9,19,11)(7,10,20,13)(8,17,12,24)(14,29,26,32)(16,30,21,31)(22,23,25,28);(1,4)(2,8)(3,12)(5,18)(6,21)(7,24)(9,26)(10,15)(11,14)(13,27)(16,19)(17,20)(22,32)(23,31)(25,29)(28,30)Polycyclic presentation:
2^9|a^2=h,b^2=i,d^2=h,e^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fghi,g^a=gi,c^b=cf,d^b=dh,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, 60384);