Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33830 |
| automorphism | Aut(G) = C26×#442 (order 4096) · Inn(G) = C22×D8 · Out(G) = C24×C4 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
340 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(6,12)(7,11)(8,16)(9,21)(13,29)(14,31)(15,28)(17,24)(18,33)(19,32)(20,26)(22,25)(27,34)(30,35);(5,6,10,15)(7,13,11,25)(8,18,20,31)(9,22,24,29)(12,17,28,21)(14,32,33,19)(16,35,26,27)(23,30,36,34);(2,4)(5,7,10,11)(6,13,15,25)(8,16,20,26)(9,17,24,21)(12,22,28,29)(14,30,33,34)(18,35,31,27)(19,23,32,36);(5,8,10,20)(6,14,15,33)(7,16,11,26)(9,23,24,36)(12,27,28,35)(13,30,25,34)(17,32,21,19)(18,29,31,22)Polycyclic presentation:
2^8|b^2=h,c^2=h,d^2=h,e^2=gh,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eg,g^a=gh,d^b=de,e^b=eg,g^b=gh,e^d=eg,g^d=gh43 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#33837, #7623585, #7623609, … |
| maximal normal subgroup of | C2×#33837, #7623585, #7623609, … |
| maximal quotient of | #491735, #491742, #491747, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26825);