Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #129998 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #8339 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #60647 | C2 |
| LCS | G ▷ C2×C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C4×#131 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
957 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 957 conjugacy classes of subgroups — too many to draw. Only its 120 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:
(1,2,5,10,19,11,31,29,6,12,22,30,4,9,18,27)(3,13,15,32,14,7,20,24,17,23,21,28,16,8,25,26);(1,3)(2,7)(4,14)(5,20)(6,17)(8,12)(9,23)(10,28)(11,13)(15,31)(16,19)(18,21)(22,25)(24,29)(26,27)(30,32);(1,3)(2,8)(4,16)(5,21)(6,17)(7,12)(9,23)(10,24)(11,13)(14,19)(15,22)(18,20)(25,31)(26,30)(27,32)(28,29)Polycyclic presentation:
2^9|a^2=f,d^2=h,e^2=i,f^2=ghi,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,h^a=hi,d^b=dh,f^b=fghi,g^b=gi,e^c=ei,f^c=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72214383, #72214424, #72214690, … |
| maximal normal subgroup of | #72214383, #72214424, #72214690, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 37195);