Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #152674 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13082 |
| commutator | C22×Q8 | C22×C4 |
| Frattini, cosocle | C2×#124 | C23 |
| prime radical | #23910 | C2 |
| LCS | G ▷ C22×Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C23×C4 ◁ C22×#124 ◁ G | |
| LECS | G ▷ C2×#124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
650 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 650 conjugacy classes of subgroups — too many to draw. Only its 66 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)(3,13,15,27)(4,12,6,9)(5,20,14,28)(7,11)(8,16,25,30)(10,19)(17,29,18,32)(21,26,31,24)(22,23);(1,3,14,30)(2,8,23,32,11,26,10,27)(4,15,19,31)(5,21,7,17)(6,18,22,16)(9,24,20,13,12,25,28,29);(1,4,7,6)(2,9,11,12)(3,15,17,18)(5,22,14,19)(8,24,26,25)(10,28,23,20)(13,27,29,32)(16,30,31,21)Polycyclic presentation:
2^9|a^2=f,b^2=dfh,c^2=i,d^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gh,h^a=hi,d^b=dg,e^b=ei,f^b=fg,g^b=gh,d^c=di,f^c=fi,f^d=fhi,g^d=gi,g^f=gi41 conjugacy classes — view the full character table.
| maximal subgroup of | #72213534, #72215744, #72215753, … |
| maximal normal subgroup of | #72213534, #72215744, #72215753, … |
| maximal quotient of | #72250745, C2×#152675, #72318074, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59871);