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×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×C22⋊C4 |
| commutator | C2×C4 | C22×C4 |
| Frattini, cosocle | C22×C4 | C23 |
| prime radical | #532 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
132 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,17,19,14)(12,15)(13,16)(18,23,20,22)(21,24);(2,6)(3,4)(7,8)(9,11)(10,14)(12,20)(13,19)(15,23)(16,17)(18,21)(22,24);(1,3,5,4)(2,6,8,7)(9,12,13,21)(10,15,16,24)(11,18,19,20)(14,22,17,23)Polycyclic presentation:
2^7|a^2=e,c^2=f,d^2=g,b^a=bd,c^a=cg,d^a=df,c^b=cf,d^b=dg,e^b=efg32 conjugacy classes — view the full character table.
| maximal subgroup of | #8428, #8429, #8430, … |
| maximal normal subgroup of | #8428, #8429, #8430, … |
| maximal quotient of | #7574, #7587, #7588, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 249);