Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | Aut(G) = C27×#120 (order 4096) · Inn(G) = C25 · Out(G) = C24×D4 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C4 | C25 |
| commutator, socle | C22 | C24×C4 |
| prime radical | D4×#41 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1068 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(9,10,12,16,13,14,19,21)(11,15,18,22,17,20,23,24);(1,3)(2,5)(4,8)(6,7);(3,8)(5,7)(10,14)(15,20)(16,21)(22,24);(2,6)(5,7)(9,11)(10,15)(12,18)(13,17)(14,20)(16,22)(19,23)(21,24);(11,17)(15,20)(18,23)(22,24)Polycyclic presentation:
2^8|a^2=g,g^2=h,b^a=bf,c^a=ch,d^a=df,c^b=cf,e^d=eh76 conjugacy classes — view the full character table.
| maximal subgroup of | #6399614, #6399615, #6399616, … |
| maximal normal subgroup of | #6399614, #6399615, #6399616, … |
| maximal quotient of | #6370183, #6370184, #6370225, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53228);