Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2812 |
| automorphism | Aut(G) = C213 (order 8192) · Inn(G) = C24 · Out(G) = C29 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C24 |
| commutator, socle | C23 | C23×C4 |
| prime radical | #392 | C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
388 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,5)(3,7)(9,10)(11,13)(12,14)(15,16)(17,18,21,24,20,23,28,31)(19,25,27,32,26,22,29,30);(1,2)(3,8)(4,5)(6,7)(9,11)(10,13)(12,15)(14,16);(2,5)(6,8)(10,14)(11,15)(17,19)(18,22)(20,26)(21,27)(23,25)(24,30)(28,29)(31,32);(1,3,4,7)(2,6,5,8)(10,14)(13,16)(19,26)(22,25)(27,29)(30,32)Polycyclic presentation:
2^8|a^2=g,d^2=e,g^2=h,b^a=be,c^a=cf,d^a=defh,c^b=cefh,d^b=de,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#23335, #6407506, #6407508, … |
| maximal normal subgroup of | C2×#23335, #6407506, #6407508, … |
| maximal quotient of | #357300, #357345, #357346, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16323);