Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3290 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×H(5,𝔽2) | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
508 conjugacy classes of subgroups; only the 167 normal subgroups are shown.
This group has 508 conjugacy classes of subgroups — too many to draw. Only its 167 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:
(1,2,4,3)(5,6)(7,18)(8,14)(9,16)(10,15)(11,17)(12,21)(13,23)(19,27)(20,34)(22,33)(24,32)(25,29)(26,30)(28,36)(31,35);(1,3,4,2)(5,7)(6,12)(8,19)(9,25)(10,21)(11,22)(13,27)(14,31)(15,33)(16,18)(17,29)(20,23)(24,36)(26,35)(28,30)(32,34);(5,8,11,24)(6,13,17,30)(7,19,22,36)(9,23,10,26)(12,27,29,28)(14,16,32,15)(18,34,33,31)(20,21,35,25);(5,8)(6,14)(7,20)(9,26)(10,23)(11,24)(12,28)(13,15)(16,30)(17,32)(18,34)(19,25)(21,36)(22,35)(27,29)(31,33)Polycyclic presentation:
2^8|a^2=g,b^2=g,c^2=h,e^2=h,f^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,d^b=de,e^b=eh,d^c=dh,f^c=fh,e^d=eh58 conjugacy classes — view the full character table.
| maximal subgroup of | #152568, C2×#22896, #6381471, … |
| maximal normal subgroup of | #152568, C2×#22896, #6381471, … |
| maximal quotient of | #226940, #226948, #226956, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15884);