Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | Aut(G) = #72234537 (order 8192) · Inn(G) = C24 · Out(G) = #6342371 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
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 | |
176 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)(3,7)(4,5)(6,8)(9,10,14,16)(11,13,12,15)(17,18,21,23)(19,20,22,24);(2,5)(6,7)(9,11,14,12)(10,15,16,13)(17,19)(18,20)(21,22)(23,24);(1,3,4,8)(2,6,5,7)(9,12,14,11)(10,15,16,13)(17,20)(18,22)(19,23)(21,24);(1,2)(3,6)(4,5)(7,8)(9,13,14,15)(10,11,16,12)Polycyclic presentation:
2^7|a^2=efg,b^2=ef,c^2=e,d^2=ef,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de32 conjugacy classes — view the full character table.
| maximal subgroup of | #12143, #12237, #36762, … |
| maximal normal subgroup of | #12143, #12237, #36762, … |
| maximal quotient of | #14400, #14432, #14462, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1521);