Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3057 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#540 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
599 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(14,18)(16,19)(20,24)(21,25)(23,26)(27,28);(2,4)(13,14,19,25,17,21,16,18)(15,20,26,28,22,27,23,24);(6,9)(10,12)(13,15,17,22)(14,20,21,27)(16,23,19,26)(18,24,25,28);(5,7,8,11)(6,10,9,12)(15,22)(20,27)(23,26)(24,28)Polycyclic presentation:
2^8|a^2=g,b^2=egh,c^2=h,d^2=f,e^2=h,b^a=be,c^a=cf,e^a=eh,d^c=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #124337, #124350, #124708, … |
| maximal normal subgroup of | #124337, #124350, #124708, … |
| maximal quotient of | #207554, #207569, #207584, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14402);