Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3019 |
| automorphism | Aut(G) = #72236986 (order 2048) · Inn(G) = C2×#120 · Out(G) = #120 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#120 |
| commutator, Frattini, cosocle | C2×C4 | C24 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
349 conjugacy classes of subgroups; only the 106 normal subgroups are shown.
This group has 349 conjugacy classes of subgroups — too many to draw. Only its 106 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)(3,11)(4,8)(5,17)(6,9)(7,13)(10,20)(12,28)(14,25)(15,26)(16,23)(18,22)(19,24)(21,29)(27,30)(31,32);(1,3)(2,7)(4,12)(5,18)(6,14)(8,21)(9,22)(10,25)(11,17)(13,20)(15,31)(16,30)(19,29)(23,32)(24,27)(26,28);(1,4,6,16)(2,8,9,23)(3,12,14,30)(5,19,20,15)(7,21,22,32)(10,26,17,24)(11,27,25,28)(13,31,18,29);(2,9)(7,22)(8,23)(10,17)(11,25)(21,32)(24,26)(27,28)Polycyclic presentation:
2^7|c^2=g,e^2=g,b^a=be,c^a=cf,d^a=dg,e^a=eg,e^b=eg,f^b=fg,e^c=eg29 conjugacy classes — view the full character table.
| maximal subgroup of | #23874, #23875, #23893, … |
| maximal normal subgroup of | #23874, #23875, #23893, … |
| maximal quotient of | #18134, #18139, #18142, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1752);