Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | Aut(G) order 24576 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
402 conjugacy classes of subgroups.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 12 points:
(1,2)(3,4)(6,8);(2,3)(10,12);(5,6)(7,8)(9,10)(11,12);(1,3,4,2)(5,6)(7,8)Polycyclic presentation:
2^7|d^2=e,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de32 conjugacy classes — view the full character table.
| Fitting subgroup of | #2675⋊C3, #11711397, #11711429 |
| maximal subgroup of | #12148, #12242, #36706, … |
| maximal normal subgroup of | #12148, #12242, #36706, … |
| maximal quotient of | #14361, #14402, #14521, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1407);