Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3003 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24 |
| prime radical | C2×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
192 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,6)(3,7,8,5)(9,10,13,15)(11,19,21,24)(12,18,14,17)(16,22,23,20);(1,3,4,8)(2,5,6,7)(10,15)(16,23)(17,18)(19,24);(9,11)(10,16)(12,20)(13,21)(14,22)(15,23)(17,19)(18,24);(9,12,13,14)(10,17,15,18)(11,20,21,22)(16,19,23,24)Polycyclic presentation:
2^7|a^2=e,b^2=eg,d^2=g,f^2=g,b^a=be,c^a=cf,d^a=dg,f^a=fg,f^c=fg32 conjugacy classes — view the full character table.
| maximal subgroup of | #9351, #9352, #24724, … |
| maximal normal subgroup of | #9351, #9352, #24724, … |
| maximal quotient of | #20649, #20875, #21034, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1812);