Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #153119 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2199 |
| commutator, Frattini, cosocle | #532 | C23 |
| socle | C2 | #13419 |
| prime radical | #31076 | C2 |
| LCS | G ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ G | |
| LECS | G ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
423 conjugacy classes of subgroups; only the 46 normal subgroups are shown.
This group has 423 conjugacy classes of subgroups — too many to draw. Only its 46 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)(3,13)(4,18)(5,15)(6,11)(7,10)(8,12)(9,21)(14,25)(16,24)(17,26)(19,23)(20,28)(22,29)(27,30)(31,32);(1,3)(2,6)(4,19)(5,16)(7,11)(8,17)(9,22)(10,26)(12,24)(13,15)(14,21)(18,32)(20,27)(23,28)(25,31)(29,30);(1,4,5,20,8,22,7,25)(2,9,10,27,12,19,15,31)(3,14,11,29,17,28,16,18)(6,23,13,30,24,21,26,32)Polycyclic presentation:
2^9|c^2=d,d^2=i,g^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fghi,h^a=hi,c^b=cf,d^b=di,e^b=eg,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^c=gh,f^e=fhi,g^e=gi,g^f=gi41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60476);