Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33551 |
| automorphism | Aut(G) = C2×#6628896 (order 1024) · Inn(G) = C2×B(4,𝔽2) · Out(G) = C23 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
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 ◁ C23 ◁ C22×D4 ◁ G | |
| LECS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
578 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 578 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) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,11)(4,8)(5,9)(6,10)(7,14)(12,24)(13,25)(15,21)(16,22)(17,30)(18,23)(19,27)(20,28)(26,32)(29,31);(1,3)(2,7)(4,12)(5,14)(6,13)(8,19)(9,11)(10,20)(15,27)(16,26)(17,29)(18,28)(21,24)(22,31)(23,25)(30,32);(1,4,6,16)(2,8,10,22)(3,12,13,26)(5,17,18,15)(7,19,20,31)(9,21,23,30)(11,24,25,32)(14,29,28,27);(3,13)(7,20)(11,25)(12,26)(14,28)(19,31)(24,32)(27,29)Polycyclic presentation:
2^8|c^2=h,b^a=be,c^a=cf,g^a=gh,d^b=dh,f^b=fgh,e^c=eg,f^e=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | #511791, #511792, #511793, … |
| maximal normal subgroup of | #511791, #511792, #511793, … |
| maximal quotient of | #484158, #484163, #484182, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26540);