Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2812 |
| automorphism | Aut(G) order 24576 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C24 |
| prime radical | C22×D4 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
228 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6,8,10)(7,9,11,12)(13,14,17,18)(15,16,19,20);(2,4)(7,11)(9,12)(14,18)(16,20);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,15)(14,16)(17,19)(18,20);(5,7,8,11)(6,9,10,12)(13,16)(14,19)(15,18)(17,20)Polycyclic presentation:
2^7|a^2=ef,d^2=f,b^a=be,c^a=cf,c^b=cg,d^b=def,d^c=de26 conjugacy classes — view the full character table.
| Fitting subgroup of | #2823⋊C3, #11711398, #11711434 |
| maximal subgroup of | #12155, #12249, C2×#2823, … |
| maximal normal subgroup of | #12155, #12249, C2×#2823, … |
| maximal quotient of | #14632, #14633, #14634, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1555);