Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2812 |
| automorphism | Aut(G) order 49152 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
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 | |
178 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,6)(3,5,7,8)(9,10,13,12)(11,14,15,16);(3,7)(5,8)(10,12)(14,16)(17,18,20,22)(19,23,24,21);(1,2)(3,5)(4,6)(7,8)(9,11)(10,14)(12,16)(13,15)(17,19,20,24)(18,21,22,23);(1,3)(2,5)(4,7)(6,8)(9,12,13,10)(11,14,15,16)Polycyclic presentation:
2^7|a^2=ef,b^2=g,c^2=g,d^2=e,b^a=be,c^a=cf,c^b=cg,d^b=def,d^c=de26 conjugacy classes — view the full character table.
| derived subgroup of | #2844⋊C3, #10358⋊C3, #10583⋊C3, … |
| Fitting subgroup of | #2844⋊C3, #11714252, #11714253, … |
| maximal subgroup of | #10358, #10359, #10583, … |
| maximal normal subgroup of | #10358, #10359, #10583, … |
| maximal quotient of | #14638, #14639, #14677, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1576);