Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29736 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C43 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#496 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
512 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 512 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) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6)(7,10)(8,9)(11,12)(13,14,19,25,18,16,17,21)(15,20,26,28,23,22,24,27);(1,2,3,4)(6,9)(10,12)(13,15)(14,20)(16,22)(17,24)(18,23)(19,26)(21,27)(25,28);(2,4)(7,11)(10,12)(13,16,18,14)(15,22,23,20)(17,21,19,25)(24,27,26,28);(5,7,8,11)(6,10,9,12)(15,23)(20,22)(24,26)(27,28)Polycyclic presentation:
2^8|a^2=fh,b^2=g,c^2=h,d^2=e,f^2=h,b^a=be,c^a=cf,c^b=cg,d^b=dh,d^c=de,f^c=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #300494, #300495, #300530, … |
| maximal normal subgroup of | #300494, #300495, #300530, … |
| maximal quotient of | #331346, #331352, #331357, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23261);