Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #353 |
| automorphism | Aut(G) = #72218077 (order 1024) · Inn(G) = #99 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #99 |
| commutator | C2×C4 | C2×C8 |
| Frattini, cosocle | C2×C4⋊C4 | C22 |
| prime radical | C23 | C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ G | |
| LECS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
59 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,2,3,5,4,6,7,8)(9,10)(11,17,19,23)(12,20,21,15)(13,16)(14,18)(22,24);(9,11)(10,14,16,22)(12,18)(13,19)(15,17,20,23)(21,24)Polycyclic presentation:
2^7|a^2=d,b^2=e,c^2=g,d^2=f,b^a=bc,c^a=ce,e^a=eg,d^b=deg,d^c=dg26 conjugacy classes — view the full character table.
| maximal subgroup of | #7352, #7353, #7359, … |
| maximal normal subgroup of | #7352, #7353, #7359, … |
| maximal quotient of | #7062, #7065, #7122, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 56);