Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453 |
| automorphism | Aut(G) = C22×#7622880 (order 2048) · Inn(G) = #120 · Out(G) = C26 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #120 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C2 | C4×#120 |
| prime radical | #3018 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
645 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,8,11,7,12,6,22)(3,13,16,9,17,26,23,27)(4,18,19,21,20,10,24,5)(14,28,30,25,31,15,29,32);(1,3)(2,9)(4,14)(5,15)(6,23)(7,17)(8,16)(10,25)(11,26)(12,27)(13,22)(18,32)(19,30)(20,31)(21,28)(24,29);(1,4)(2,10)(3,14)(5,11)(6,24)(7,20)(8,19)(9,25)(12,18)(13,28)(15,26)(16,30)(17,31)(21,22)(23,29)(27,32);(1,5,8,18,7,21,6,10)(2,4,11,19,12,20,22,24)(3,15,16,32,17,28,23,25)(9,14,26,30,27,31,13,29)Polycyclic presentation:
2^8|a^2=g,d^2=g,e^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,e^b=eh,f^b=fh,e^c=eh64 conjugacy classes — view the full character table.
| maximal subgroup of | #136831, #136832, #136833, … |
| maximal normal subgroup of | #136831, #136832, #136833, … |
| maximal quotient of | C8×#453, C8×#455, C8×#454, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14465);