Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2812 |
| automorphism | Aut(G) order 24576 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
356 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(6,9)(10,12)(13,14)(15,18)(16,17)(19,20);(2,4)(6,9)(7,11)(14,17)(15,19);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,20)(16,17)(18,19);(5,7)(6,10)(8,11)(9,12)(13,15)(14,18)(16,19)(17,20)Polycyclic presentation:
2^7|b^a=be,c^a=cf,c^b=cg,d^b=def,d^c=de26 conjugacy classes — view the full character table.
| Fitting subgroup of | #2812⋊C3, #11711386, #11711404 |
| maximal subgroup of | #12093, #12205, C2×#2812, … |
| maximal normal subgroup of | #12093, #12205, C2×#2812, … |
| maximal quotient of | #14621, #14622, #14627, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1544);