Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23860 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
374 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 374 conjugacy classes of subgroups — too many to draw. Only its 127 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) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,3,4)(5,6,8,9)(7,11,12,10);(6,9)(10,11)(13,14,18,21)(15,22,25,28)(16,20,17,26)(19,23,27,24);(2,4)(13,15,18,25)(14,19,21,27)(16,24,17,23)(20,28,26,22);(5,7,8,12)(6,10,9,11)(13,16,18,17)(14,20,21,26)(15,23,25,24)(19,22,27,28)Polycyclic presentation:
2^8|a^2=ef,b^2=h,c^2=h,d^2=eh,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gh,d^c=dh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #273444, #273459, #273480, … |
| maximal normal subgroup of | #273444, #273459, #273480, … |
| maximal quotient of | #276527, #276528, #276529, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17686);