Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33359 |
| automorphism | Aut(G) order 24576 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C22×#99 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C2×D4 | C24 |
| prime radical | #3499 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×Q8 ◁ C2×2−1+4 ◁ G | |
| LECS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
594 conjugacy classes of subgroups; only the 181 normal subgroups are shown.
This group has 594 conjugacy classes of subgroups — too many to draw. Only its 181 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,8)(4,9)(5,10,18,7)(6,11)(12,21)(13,22,28,19)(14,23)(15,24)(16,25,30,20)(17,26)(27,32,29,31);(2,7)(5,18)(8,19)(9,20)(10,11)(13,28)(16,30)(21,31)(22,23)(24,32)(25,26)(27,29);(1,3,6,14)(2,8,11,23)(4,15,17,12)(5,13,18,28)(7,19,10,22)(9,24,26,21)(16,29,30,27)(20,32,25,31);(1,4,6,17)(2,9,11,26)(3,12,14,15)(5,16,18,30)(7,20,10,25)(8,21,23,24)(13,27,28,29)(19,31,22,32)Polycyclic presentation:
2^8|a^2=f,c^2=h,d^2=h,e^2=h,b^a=be,e^a=eg,g^a=gh,e^b=eh,f^b=fgh,d^c=dh,f^e=fh46 conjugacy classes — view the full character table.
| Fitting subgroup of | #33360⋊C3 |
| maximal subgroup of | #151823, #152703, #152747, … |
| maximal normal subgroup of | #151823, #152703, #152747, … |
| maximal quotient of | #480015, #480049, #480121, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26348);