Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C22 | C25 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
502 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,5)(3,7)(10,14)(12,20)(15,17)(18,24);(2,5)(6,8)(9,10,13,14)(11,17,19,15)(12,16,20,21)(18,23,24,22);(1,2,4,5)(3,6,7,8);(1,3)(2,6)(4,7)(5,8)(9,11,13,19)(10,15,14,17)(12,18,20,24)(16,22,21,23);(9,12)(10,16)(11,18)(13,20)(14,21)(15,22)(17,23)(19,24)Polycyclic presentation:
2^7|b^2=f,c^2=g,d^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 | #8498, #12782, #18684, … |
| maximal normal subgroup of | #8498, #12782, #18684, … |
| maximal quotient of | #34365, #34384, #34409, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 2276);