Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23924 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
362 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 362 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) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,7)(4,5)(6,8)(9,10)(11,15)(12,14)(13,16)(17,18,23,27)(19,24,20,29)(21,26,22,31)(25,28,32,30);(2,5)(6,7)(9,11)(10,13)(12,16)(14,15)(17,19,23,20)(18,24,27,29)(21,30,22,28)(25,31,32,26);(11,16)(13,15)(17,20,23,19)(18,25,27,32)(21,30,22,28)(24,31,29,26);(1,3,4,8)(2,6,5,7)(10,14)(13,15)(17,21,23,22)(18,26,27,31)(19,28,20,30)(24,32,29,25)Polycyclic presentation:
2^8|a^2=h,b^2=h,c^2=h,d^2=egh,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,d^b=dh,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #123020, #123030, C2×#29427, … |
| maximal normal subgroup of | #123020, #123030, C2×#29427, … |
| maximal quotient of | #363418, #371110, #371148, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 22415);