Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C23 |
| commutator, socle | C22 | C23×C4 |
| Frattini, cosocle | C2×C4 | C24 |
| prime radical | C23 | C22×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
182 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)(3,8)(4,6)(5,7)(9,10,12,16,13,15,19,22)(11,14,17,21,18,20,23,24);(1,3,5,4)(2,6,7,8);(1,4,5,3)(2,6,7,8)(9,11,12,17,13,18,19,23)(10,14,16,21,15,20,22,24);(10,15)(14,20)(16,22)(21,24)Polycyclic presentation:
2^7|a^2=f,b^2=e,c^2=ef,f^2=g,b^a=be,d^a=dg56 conjugacy classes — view the full character table.
| Frattini subgroup of | #94265, #94266, #94267, … |
| maximal subgroup of | #8242, #8244, #8246, … |
| maximal normal subgroup of | #8242, #8244, #8246, … |
| maximal quotient of | #20435, #20437, #20516, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1692);