Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1468 |
| automorphism | Aut(G) = #72232993 (order 2048) · Inn(G) = C2×C22⋊C4 · Out(G) = C23×D4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×C22⋊C4 |
| commutator | C23 | C22×C4 |
| Frattini, cosocle | C22×C4 | C23 |
| prime radical | C2×#121 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
161 conjugacy classes of subgroups; only the 48 normal subgroups are shown.
This group has 161 conjugacy classes of subgroups — too many to draw. Only its 48 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2)(3,7,8,5)(4,6)(9,10,12,16,13,14,11,15);(1,3)(2,5)(4,8)(6,7)(10,14)(11,12);(9,11,13,12)(10,15,14,16)Polycyclic presentation:
2^7|a^2=e,c^2=g,e^2=g,b^a=bd,d^a=df,c^b=cg,e^b=ef32 conjugacy classes — view the full character table.
| maximal subgroup of | #8465, #8466, #8536, … |
| maximal normal subgroup of | #8465, #8466, #8536, … |
| maximal quotient of | #7664, #7667, #7670, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 203);