Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11704 |
| 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 | C2×#102 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | #3050 | C2 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
222 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 222 conjugacy classes of subgroups — too many to draw. Only its 63 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,7,8,5)(4,6)(9,10,13,18,15,20,14,19)(11,21,24,34,22,37,25,38)(12,26,28,17,30,35,29,36)(16,27,32,23,31,40,33,39);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,27)(13,25)(14,24)(15,22)(17,34)(18,33)(19,32)(20,31)(21,35)(23,29)(26,37)(28,39)(30,40)(36,38);(11,22)(16,31)(21,37)(23,39)(24,25)(27,40)(32,33)(34,38)Polycyclic presentation:
2^8|a^2=e,d^2=gh,e^2=h,f^2=h,g^2=h,b^a=bd,d^a=df,f^a=fh,g^a=gh,c^b=ch,d^b=dg,e^b=efgh,f^b=fh,g^b=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #128811, #128839, #129768, … |
| maximal normal subgroup of | #128811, #128839, #129768, … |
| maximal quotient of | #122584, #122587, #122592, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4698);