Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #93553 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1300 |
| commutator | C2×C42 | C42 |
| Frattini, cosocle | C4×#97 | C22 |
| socle | C2 | #7043 |
| prime radical | C23 | #344 |
| LCS | G ▷ C2×C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C42 ◁ C4×#97 ◁ G | |
| LECS | G ▷ C4×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
161 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 161 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 — 2 generators on 32 points:
(1,2,4,10,5,11,19,27,7,12,18,28,6,9,20,25)(3,8,14,21,13,23,17,31,16,24,30,32,15,22,29,26);(1,3)(2,8,11,22,12,24,9,23)(4,17,18,29)(5,15)(6,13)(7,16)(10,26,25,21,28,31,27,32)(14,20,30,19)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fh,d^2=f,e^2=fg,f^2=i,g^2=i,h^2=i,b^a=bc,c^a=cf,e^a=efgh,g^a=gi,h^a=hi,c^b=cg,d^b=dh,f^b=fi,g^b=gi,d^c=di,e^d=ei56 conjugacy classes — view the full character table.
| maximal subgroup of | #72257876, #72265486, #72265493 |
| maximal normal subgroup of | #72257876, #72265486, #72265493 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 817);