Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #58990 |
| automorphism | Aut(G) order 196608 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
2938 conjugacy classes of subgroups.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 12 points:
(2,4)(10,12);(1,2)(3,4)(6,8)(10,12);(9,10)(11,12);(5,6)(7,8)(9,10)(11,12);(1,2)(3,4)Polycyclic presentation:
2^8|b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh64 conjugacy classes — view the full character table.
| Fitting subgroup of | #58990⋊C3, #72240889, #72340903 |
| maximal subgroup of | #133390, #499787, #500218, … |
| maximal normal subgroup of | #133390, #499787, #500218, … |
| maximal quotient of | #531607, #532352, #533018, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 51978);