Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #153625 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13673 |
| commutator, Frattini, cosocle | #492 | C23 |
| prime radical | #12314 | C2 |
| LCS | G ▷ #492 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C42 ◁ #492 ◁ G | |
| LECS | G ▷ #492 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
270 conjugacy classes of subgroups; only the 30 normal subgroups are shown.
This group has 270 conjugacy classes of subgroups — too many to draw. Only its 30 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)(3,12)(4,8)(5,19)(6,10)(7,14)(9,22)(11,15)(13,27)(16,28)(17,29)(18,26)(20,24)(21,23)(25,30)(31,32);(1,3,6,15)(2,7,10,24)(4,13,18,17)(5,20,22,14)(8,23,26,32)(9,11,19,12)(16,31,30,21)(25,29,28,27);(1,4,6,18)(2,8,10,26)(3,13,15,17)(5,21,22,31)(7,23,24,32)(9,28,19,25)(11,27,12,29)(14,30,20,16)Polycyclic presentation:
2^9|b^2=i,c^2=i,d^2=i,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fg,g^a=gi,h^a=hi,d^b=di,e^b=efgh,g^b=gh,d^c=df,f^c=fhi,g^c=ghi,e^d=egh,f^d=fhi,g^d=gi,f^e=fhi,g^e=gi29 conjugacy classes — view the full character table.
| maximal subgroup of | #72231248, C2×#153628, #72283015, … |
| maximal normal subgroup of | #72231248, C2×#153628, #72283015, … |
| maximal quotient of | C2×#153628, #72283566, #72309216 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60824);