Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2846 |
| automorphism | Aut(G) order 49152 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
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 | |
612 conjugacy classes of subgroups.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 12 points:
(2,4)(6,8)(10,12);(9,10,11,12);(5,6)(7,8);(1,2)(3,4)(5,6)(7,8)Polycyclic presentation:
2^7|b^2=e,b^a=be,c^a=cf,d^a=dg44 conjugacy classes — view the full character table.
| Fitting subgroup of | C4⋊(#120⋊C3), #11711262, #11711265, … |
| maximal subgroup of | #9113, #9117, #9147, … |
| maximal normal subgroup of | #9113, #9117, #9147, … |
| maximal quotient of | #14680, #14694, #14696, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1579);