Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9980 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #392 |
| commutator, Frattini, cosocle | C23×C4 | C23 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
251 conjugacy classes of subgroups; only the 49 normal subgroups are shown.
This group has 251 conjugacy classes of subgroups — too many to draw. Only its 49 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) = 64 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2)(3,6)(4,7)(5,8)(9,12)(10,13)(11,15)(14,16)(17,18,23,28)(19,29,32,40)(20,25,36,43)(21,27,22,26)(24,33,39,31)(30,45,46,41)(34,48,47,38)(35,44,37,42);(1,3,5,10)(2,6,8,13)(4,9,11,16)(7,14,15,12)(17,19)(18,24)(20,34)(21,33)(22,31)(23,32)(25,41)(26,40)(27,29)(28,39)(30,35)(36,47)(37,46)(38,42)(43,45)(44,48);(1,4)(2,7)(3,9)(5,11)(6,12)(8,15)(10,16)(13,14)(17,20,23,36)(18,25,28,43)(19,30,32,46)(21,35,22,37)(24,38,39,48)(26,42,27,44)(29,41,40,45)(31,34,33,47)Polycyclic presentation:
2^8|a^2=g,b^2=h,c^2=g,d^2=g,f^2=g,b^a=bd,c^a=ce,d^a=dg,f^a=fgh,c^b=cf,d^b=dg,e^b=eg,f^b=fg,d^c=dh,f^c=fg28 conjugacy classes — view the full character table.
| maximal subgroup of | #151345, C2×#9985, #232512, … |
| maximal normal subgroup of | #151345, C2×#9985, #232512, … |
| maximal quotient of | #102478, #102492, #102494, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2973);