Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1510 |
| automorphism | Aut(G) = #72303605 (order 4096) · Inn(G) = C2×C22⋊C4 · Out(G) = C2×#392 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×C22⋊C4 |
| commutator | C2×C4 | C22×C8 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C22 | C2×C22⋊C8 |
| prime radical | C2×#41 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
153 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 153 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) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2)(3,9,11,6)(4,7)(5,8)(10,15,16,13)(12,14)(17,18,21,27,22,28,36,45,23,25,37,41,20,26,34,43)(19,29,31,46,30,39,47,44,32,42,35,48,33,24,38,40);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,19)(18,24)(20,33)(21,35)(22,30)(23,32)(25,39)(26,42)(27,44)(28,29)(31,37)(34,47)(36,38)(40,41)(43,46)(45,48);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(18,25)(24,39)(26,28)(27,41)(29,42)(40,44)(43,45)(46,48)Polycyclic presentation:
2^8|a^2=e,c^2=f,d^2=h,e^2=g,g^2=h,b^a=bd,c^a=ch,d^a=df,d^b=dh,e^b=efh52 conjugacy classes — view the full character table.
| maximal subgroup of | #128442, #129689, #129698, … |
| maximal normal subgroup of | #128442, #129689, #129698, … |
| maximal quotient of | #124101, #124132, #124293, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4451);