Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×D8 |
| automorphism | Aut(G) order 6144 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×Q8 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
168 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,5)(6,7)(9,10,13,16)(11,17,19,24)(12,20,22,15)(14,21,23,18);(1,2,4,5)(3,7,8,6);(9,11,13,19)(10,14,16,23)(12,21,22,18)(15,24,20,17);(1,3,4,8)(2,6,5,7)Polycyclic presentation:
2^7|a^2=g,b^2=e,c^2=g,d^2=e,f^2=g,b^a=be,c^a=cf,f^a=fg,d^b=de,f^c=fg35 conjugacy classes — view the full character table.
| Fitting subgroup of | #3363⋊C3, #11712314, #11712318, … |
| maximal subgroup of | #19812, #19816, #19817, … |
| maximal normal subgroup of | #19812, #19816, #19817, … |
| maximal quotient of | Q8×#103, #22542, #22740, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 2095);