Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2613 |
| automorphism | Aut(G) order 98304 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C24 |
| prime radical | C22×D4 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
391 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(6,8)(9,10,13,16)(11,17,19,14)(12,15,21,20)(18,23,24,22);(2,4)(6,8)(9,11,13,19)(10,14,16,17)(12,18,21,24)(15,22,20,23);(1,2)(3,4)(5,6)(7,8)(9,10)(11,17)(12,20)(13,16)(14,19)(15,21)(18,22)(23,24);(9,12)(10,15)(11,18)(13,21)(14,22)(16,20)(17,23)(19,24)Polycyclic presentation:
2^7|a^2=e,b^2=e,b^a=be,c^a=cf,c^b=cg,d^c=de32 conjugacy classes — view the full character table.
| Fitting subgroup of | #2614⋊C3, #11711946, #11711947 |
| maximal subgroup of | #18225, #18227, #23905, … |
| maximal normal subgroup of | #18225, #18227, #23905, … |
| maximal quotient of | #14256, #14263, #14301, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1346);