Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×D4 |
| commutator, Frattini, cosocle | C22×C4 | C23 |
| prime radical | C22×D4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
195 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 195 conjugacy classes of subgroups — too many to draw. Only its 54 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 — 3 generators on 24 points:
(1,2)(3,4)(9,10,14,18)(11,19,22,23)(12,17,13,16)(15,21,24,20);(2,4)(5,6)(7,8)(9,11,14,22)(10,15,18,24)(12,21,13,20)(16,23,17,19);(6,8)(10,16)(12,13)(15,23)(17,18)(19,24)(20,21)Polycyclic presentation:
2^7|a^2=g,b^2=g,d^2=g,e^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eg,c^b=cf,d^b=dg,e^b=eg,d^c=dg,e^c=eg32 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#2015, #23421, #23423, … |
| maximal normal subgroup of | C2×#2015, #23421, #23423, … |
| maximal quotient of | #9955, #9956, #9958, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 747);