Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2846 |
| automorphism | Aut(G) order 98304 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
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 | |
462 conjugacy classes of subgroups.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 12 points:
(2,4)(6,8)(10,11);(9,10,12,11);(5,6,7,8)(9,11,12,10);(1,2)(3,4)Polycyclic presentation:
2^7|b^2=e,c^2=f,b^a=be,c^a=cf,d^a=dg44 conjugacy classes — view the full character table.
| Fitting subgroup of | #2856⋊C3, #11713414, #11713613 |
| maximal subgroup of | #7664, #7684, #7704, … |
| maximal normal subgroup of | #7664, #7684, #7704, … |
| maximal quotient of | #14686, #14698, #14700, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1588);