Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499825 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #23900 |
| commutator, Frattini, cosocle | C2×#41 | C24 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×2−1+4 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
1872 conjugacy classes of subgroups; only the 148 normal subgroups are shown.
This group has 1872 conjugacy classes of subgroups — too many to draw. Only its 148 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 — 4 generators on 32 points:
(1,2)(3,13)(4,11)(5,10)(6,21)(7,12)(8,16)(9,18)(14,29)(15,23)(17,28)(19,27)(20,26)(22,31)(24,30)(25,32);(1,3)(2,8)(4,15)(5,16)(6,14)(7,17)(9,23)(10,13)(11,22)(12,25)(18,31)(19,32)(20,30)(21,24)(26,29)(27,28);(3,14)(4,6)(5,19)(8,22)(9,26)(11,21)(13,23)(15,17)(16,31)(24,25)(28,29)(30,32);(1,4,7,6)(2,9,12,26)(3,15,17,14)(5,18,19,20)(8,23,25,29)(10,11,27,21)(13,22,28,24)(16,31,32,30)Polycyclic presentation:
2^9|d^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,h^a=hi,c^b=cg,f^b=fh,g^b=gi,d^c=di,e^c=eh,g^c=gi,f^d=fhi,f^e=fi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72236805, #72236891, #72236997, … |
| maximal normal subgroup of | #72236997, #72237521, #72237529, … |
| maximal quotient of | #72311943, #72312426, #72337433, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407023);