Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2202 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | B(4,𝔽2) |
| commutator | C2×Q8 | C22×C4 |
| Frattini, cosocle | C22×Q8 | C23 |
| prime radical | C23 | #120 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
212 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 212 conjugacy classes of subgroups — too many to draw. Only its 59 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 — 3 generators on 36 points:
(1,2,4,3)(5,6)(7,17)(8,14)(9,22)(10,25)(11,16)(12,18)(13,21)(15,27)(19,35)(20,32)(23,30)(24,31)(26,28)(29,36)(33,34);(1,3,4,2)(5,7)(6,12)(8,18)(9,23)(10,26)(11,20)(13,28)(14,32)(15,29)(16,30)(17,22)(19,27)(21,36)(24,34)(25,33)(31,35);(5,8,11,9)(6,13,16,31)(7,18,20,23)(10,24,27,21)(12,28,30,35)(14,15,22,25)(17,33,32,29)(19,36,26,34)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=h,d^2=h,e^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gh,d^b=dh,e^b=egh,d^c=dg,e^c=eh,e^d=eh34 conjugacy classes — view the full character table.
| maximal subgroup of | #152480, #153306, C2×#12971, … |
| maximal normal subgroup of | #152480, #153306, C2×#12971, … |
| maximal quotient of | #135943, #135963, #135977, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5959);