Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12173 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1598 |
| commutator, Frattini, cosocle | C2×M16 | C24 |
| socle | C2 | C2×#1598 |
| LCS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C22×M16 ◁ G | |
| LECS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
907 conjugacy classes of subgroups; only the 136 normal subgroups are shown.
This group has 907 conjugacy classes of subgroups — too many to draw. Only its 136 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)(4,9)(5,19)(6,11)(7,12)(8,15)(10,21)(14,29)(16,26)(17,30)(18,28)(20,23)(22,27)(24,32)(25,31);(1,3)(2,8)(4,14)(5,20)(6,16)(7,17)(9,23)(10,26)(11,24)(12,25)(13,19)(15,21)(18,31)(22,32)(27,30)(28,29);(2,9,12,11)(8,23,25,24)(10,27,28,19)(13,26,30,29);(1,4,7,6)(2,9,12,11)(3,14,17,16)(5,18,22,21)(8,23,25,24)(10,19,28,27)(13,29,30,26)(15,20,31,32)Polycyclic presentation:
2^9|c^2=h,d^2=i,e^2=gi,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fi,g^a=gi,h^a=hi,e^b=eg,f^b=fhi,g^b=gi,e^c=eh,f^e=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72306773, #72307392, #72307417, … |
| maximal normal subgroup of | #72306773, #72307392, #72307417, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 391853);