Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C22 | C25 |
| prime radical | #532 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
501 conjugacy classes of subgroups; only the 384 normal subgroups are shown.
This group has 501 conjugacy classes of subgroups — too many to draw. Only its 384 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,5)(3,7)(9,10,14,12)(11,17,19,15)(13,20,21,16)(18,22,24,23);(2,5)(6,8)(9,11,14,19)(10,15,12,17)(13,18,21,24)(16,22,20,23);(1,2,4,5)(3,6,7,8);(1,3)(2,6)(4,7)(5,8)(9,12,14,10)(11,15,19,17)(13,20,21,16)(18,22,24,23);(9,13,14,21)(10,16,12,20)(11,18,19,24)(15,22,17,23)Polycyclic presentation:
2^7|a^2=f,b^2=f,c^2=g,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=df38 conjugacy classes — view the full character table.
| maximal subgroup of | #8492, #8496, #8531, … |
| maximal normal subgroup of | #8492, #8496, #8531, … |
| maximal quotient of | #34368, #34374, #34413, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 2282);