Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33557 |
| automorphism | Aut(G) = #72342352 (order 4096) · Inn(G) = C2×B(4,𝔽2) · Out(G) = C25 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×D4 | C24 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ G | |
| LECS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
624 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 624 conjugacy classes of subgroups — too many to draw. Only its 109 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,2)(3,7)(4,6)(5,8)(9,15)(10,12)(11,13)(14,16);(2,6)(3,9)(7,14)(8,13)(10,16)(12,15);(3,9)(4,11)(7,15)(12,14);(1,3,5,9)(2,7,8,15)(4,10,11,16)(6,12,13,14)Polycyclic presentation:
2^8|d^2=h,b^a=be,c^a=cf,g^a=gh,d^b=dh,f^b=fgh,d^c=dh,e^c=eg,f^e=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | #152567, #152573, #153121, … |
| maximal normal subgroup of | #152567, #152573, #153121, … |
| maximal quotient of | #484164, #484178, #484250, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26547);