Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2613 |
| automorphism | Aut(G) = C212 (order 4096) · Inn(G) = C24 · Out(G) = C28 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C23 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
184 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 184 conjugacy classes of subgroups — too many to draw. Only its 96 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,5)(6,8)(9,10,12,14)(11,13,16,15)(19,23)(22,24);(9,10)(11,13)(12,14)(15,16)(17,18,21,20)(19,24,23,22);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,14)(13,16)(17,19,21,23)(18,22,20,24);(1,3,4,7)(2,6,5,8)(9,11,12,16)(10,13,14,15)(17,20,21,18)(19,24,23,22)Polycyclic presentation:
2^7|a^2=e,b^2=g,c^2=g,d^2=f,b^a=be,c^a=cf,c^b=cg,d^c=de32 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#2666, #34589, #34592, … |
| maximal normal subgroup of | C2×#2666, #34589, #34592, … |
| maximal quotient of | #14284, #14285, #14286, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1398);