Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) = C22×C8×#120 (order 1024) · Inn(G) = C23 · Out(G) = C24×C8 |
| homology | Abelianization (H1) = C22×C16 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C16 | C23 |
| commutator, socle | C22 | C22×C16 |
| prime radical | C2×D4 | C16 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C16 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
101 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6,9,14,10,15,23,30,11,7,16,18,24,19,31,33,8,12,20,25,21,26,34,35,22,17,13,27,28,32,29,36);(2,4)(6,12)(7,17)(14,25)(15,26)(18,27)(19,32)(30,35)(33,36);(5,7,9,18,10,19,23,33,11,12,16,25,24,26,31,35,8,17,20,27,21,32,34,36,22,6,13,14,28,15,29,30)Polycyclic presentation:
2^8|a^2=e,c^2=e,e^2=f,f^2=g,g^2=h,b^a=bd,c^a=ch,c^b=ch112 conjugacy classes — view the full character table.
| maximal subgroup of | #94604, #151071, #151074, … |
| maximal normal subgroup of | #94604, #151071, #151074, … |
| maximal quotient of | #151071, #151090, #151092, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6634);