Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2846 |
| automorphism | Aut(G) order 24576 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C24 |
| prime radical | C22×D4 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
258 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(6,7)(10,11)(15,18)(16,20);(5,6,9,7)(8,10,12,11)(13,14,17,19)(15,16,18,20);(1,2,3,4)(5,7,9,6)(8,11,12,10)(13,15,17,18)(14,16,19,20);(5,8,9,12)(6,10,7,11)(13,16)(14,18)(15,19)(17,20)Polycyclic presentation:
2^7|b^2=eg,c^2=f,d^2=e,b^a=be,c^a=cf,d^a=dg44 conjugacy classes — view the full character table.
| Fitting subgroup of | C4⋊(#126⋊C3) |
| maximal subgroup of | #7666, #7672, #7686, … |
| maximal normal subgroup of | #7666, #7672, #7686, … |
| maximal quotient of | #14688, #14699, #14700, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1592);