Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | Aut(G) order 24576 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C43 |
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 | |
214 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,5,3)(4,7,8,6)(9,10)(11,14)(12,13)(15,16)(18,19)(22,23);(2,3)(4,8)(10,12)(14,15)(17,18,21,19)(20,23,24,22);(1,3,5,2)(4,7,8,6)(9,11,13,16)(10,14,12,15)(17,19,21,18)(20,23,24,22);(1,4,5,8)(2,6,3,7)(9,12,13,10)(11,15,16,14)(17,20)(18,22)(19,23)(21,24)Polycyclic presentation:
2^7|a^2=g,b^2=f,c^2=e,d^2=ef,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de32 conjugacy classes — view the full character table.
| Fitting subgroup of | #2785⋊C3, #11711396, #11711428 |
| maximal subgroup of | #12145, #12239, #36796, … |
| maximal normal subgroup of | #12145, #12239, #36796, … |
| maximal quotient of | #14399, #14476, #14589, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1517);