Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2613 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C24 |
| commutator, socle | C23 | C23×C4 |
| prime radical | #2397 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
501 conjugacy classes of subgroups; only the 163 normal subgroups are shown.
This group has 501 conjugacy classes of subgroups — too many to draw. Only its 163 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(5,6)(7,10)(8,11)(9,12)(13,14,17,21,18,15,19,23)(16,20,25,27,24,22,26,28);(5,7,10,6)(8,11,9,12)(13,15,17,23,18,14,19,21)(16,20,25,27,24,22,26,28);(1,2,3,4)(5,8,10,9)(6,11,7,12)(13,16)(14,20)(15,22)(17,25)(18,24)(19,26)(21,27)(23,28);(5,9,10,8)(6,11,7,12)(16,24)(20,22)(25,26)(27,28)Polycyclic presentation:
2^8|a^2=g,b^2=eg,c^2=ef,d^2=e,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=ceh,d^c=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #122418, #122426, #128966, … |
| maximal normal subgroup of | #122418, #122426, #128966, … |
| maximal quotient of | #219393, #219394, #219395, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15251);