Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C23 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
190 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,5,3)(4,7,8,6)(10,12)(14,15)(17,18,21,24)(19,23,20,22);(2,3)(6,7)(9,10,13,12)(11,14,16,15)(17,19,21,20)(18,22,24,23);(1,3,5,2)(4,7,8,6)(9,11,13,16)(10,14,12,15)(17,18)(19,22)(20,23)(21,24);(1,4,5,8)(2,6,3,7)(9,12,13,10)(11,15,16,14)(17,20,21,19)(18,23,24,22)Polycyclic presentation:
2^7|a^2=fg,b^2=eg,c^2=ef,d^2=e,b^a=be,c^a=cf,d^a=de,c^b=cg38 conjugacy classes — view the full character table.
| maximal subgroup of | #10351, #10352, #10353, … |
| maximal normal subgroup of | #10351, #10352, #10353, … |
| maximal quotient of | #13911, #13982, #14030, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1303);