Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62295 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
983 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,6,8)(3,7,5,4)(9,10)(11,18)(12,16)(13,22)(14,17)(15,20)(19,24)(21,23);(1,3,6,5)(2,7,8,4)(9,11)(10,15)(12,19)(13,21)(14,20)(16,23)(17,18)(22,24);(1,4,5,2,6,7,3,8)(9,12,14,13)(10,16,17,22)(11,19,20,21)(15,23,18,24);(1,3,6,5)(2,7,8,4)(11,20)(15,18)(19,21)(23,24);(1,3,6,5)(2,7,8,4)(9,13,14,12)(10,16,17,22)(11,21,20,19)(15,23,18,24)Polycyclic presentation:
2^8|a^2=h,b^2=h,c^2=fg,d^2=h,e^2=fh,g^2=h,b^a=bf,c^a=cg,e^a=ef,g^a=gh,c^b=ch,d^b=df,d^c=dh,e^c=eh49 conjugacy classes — view the full character table.
| Fitting subgroup of | #62595⋊C3 |
| maximal subgroup of | #250393, #250397, #250401, … |
| maximal normal subgroup of | #250393, #250397, #250401, … |
| maximal quotient of | #6386963, #6389662, #6389999, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55583);