Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1504 |
| automorphism | Aut(G) = #72232438 (order 2048) · Inn(G) = C2×C22⋊C4 · Out(G) = C23×D4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×C22⋊C4 |
| commutator | C2×C4 | C22×C4 |
| Frattini, cosocle | C42 | C23 |
| prime radical | C23 | C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
106 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,6,7,8)(9,10,12,16,13,15,21,23)(11,17,18,24,19,14,20,22);(1,2)(3,6)(4,8)(5,7)(9,11,13,19)(10,14,15,17)(12,20,21,18)(16,24,23,22);(1,3,5,4)(2,6,7,8)(10,15)(14,17)(16,23)(22,24)Polycyclic presentation:
2^7|a^2=e,b^2=f,c^2=g,d^2=g,e^2=f,b^a=bd,c^a=cf,d^a=df,d^b=dg,e^b=efg32 conjugacy classes — view the full character table.
| maximal subgroup of | #8417, #8418, #8457, … |
| maximal normal subgroup of | #8417, #8418, #8457, … |
| maximal quotient of | #7579, #7588, #7591, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 263);