Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3019 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
847 conjugacy classes of subgroups; only the 239 normal subgroups are shown.
This group has 847 conjugacy classes of subgroups — too many to draw. Only its 239 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6,10,16,11,14,9,15)(7,17,21,12,20,28,19,27)(8,22,24,29,25,13,23,30)(18,31,34,35,33,36,32,26);(1,2,3,4)(5,7)(6,12)(8,18)(9,21)(10,19)(11,20)(13,26)(14,27)(15,28)(16,17)(22,35)(23,34)(24,32)(25,33)(29,31)(30,36);(5,8,11,25)(6,13,14,22)(7,18,20,33)(9,24,10,23)(12,26,27,35)(15,30,16,29)(17,31,28,36)(19,34,21,32);(6,14)(12,27)(13,22)(15,16)(17,28)(26,35)(29,30)(31,36)Polycyclic presentation:
2^8|a^2=g,b^2=eg,c^2=h,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,f^a=fh,e^b=eh,f^b=fh,g^b=gh,e^c=eh,g^c=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | #148491, #152318, #152320, … |
| maximal normal subgroup of | #148491, #152318, #152320, … |
| maximal quotient of | #193790, #193799, #193803, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13942);