Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19014 |
| automorphism | Aut(G) = C2×#72286730 (order 4096) · Inn(G) = C2×#120 · Out(G) = C2×#120 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
454 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 454 conjugacy classes of subgroups — too many to draw. Only its 111 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) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,13)(4,15)(5,10)(6,11)(7,12)(8,14)(9,16)(18,22)(20,27)(23,31)(24,28)(25,29)(30,32);(1,3,7,4)(2,8,12,9)(5,16,6,14)(10,13,11,15)(17,18,21,24)(19,23,26,32)(20,28,27,22)(25,30,29,31);(1,4,7,3)(2,9,12,8)(5,16,6,14)(10,13,11,15)(17,19,21,26)(18,23,24,32)(20,29,27,25)(22,30,28,31);(1,5,7,6)(2,10,12,11)(3,14,4,16)(8,15,9,13)(18,24)(22,28)(23,32)(30,31)Polycyclic presentation:
2^8|b^2=g,c^2=g,d^2=gh,e^2=g,f^2=gh,b^a=be,c^a=cf,e^a=eg,f^a=fgh,d^b=dg,e^b=eg,f^b=fh,e^c=eh,f^c=fgh37 conjugacy classes — view the full character table.
| maximal subgroup of | #244548, #244549, #244555, … |
| maximal normal subgroup of | #244548, #244549, #244555, … |
| maximal quotient of | #166331, #166337, #166387, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12036);