Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | Aut(G) order 49152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
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 | |
738 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 20 points:
(2,4)(5,6,9,12)(7,10,15,13)(8,16,17,11)(14,19,20,18);(5,6)(7,13)(8,11)(9,12)(10,15)(14,19)(16,17)(18,20);(1,2,3,4);(5,7)(6,10)(8,14)(9,15)(11,18)(12,13)(16,19)(17,20);(5,8,9,17)(6,11,12,16)(7,14,15,20)(10,18,13,19)Polycyclic presentation:
2^7|a^2=f,c^2=g,e^2=f,b^a=bf,c^a=cg,e^a=ef,d^b=df44 conjugacy classes — view the full character table.
| Fitting subgroup of | #3499⋊C3, #11711042, #11711060, … |
| maximal subgroup of | #8421, #8471, #12717, … |
| maximal normal subgroup of | #8421, #8471, #12717, … |
| maximal quotient of | #34183, #34207, #34209, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 2231);