Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2613 |
| automorphism | Aut(G) order 24576 |
| 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 | |
176 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,5,3)(4,6,8,7)(11,15)(13,16)(17,18,21,19)(20,23,24,22);(2,3)(6,7)(9,10,12,14)(11,16,15,13)(17,19,21,18)(20,23,24,22);(1,3,5,2)(4,6,8,7)(9,11)(10,13)(12,15)(14,16)(17,20)(18,22)(19,23)(21,24);(1,4,5,8)(2,6,3,7)Polycyclic presentation:
2^7|a^2=fg,b^2=f,c^2=e,d^2=e,b^a=be,c^a=cf,c^b=cg,d^c=de32 conjugacy classes — view the full character table.
| Fitting subgroup of | #2645⋊C3 |
| maximal subgroup of | #29438, #29439, #29440, … |
| maximal normal subgroup of | #29438, #29439, #29440, … |
| maximal quotient of | #14261, #14266, #14333, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1377);