Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #22455 |
| automorphism | Aut(G) = C2×#72236721 (order 4096) · Inn(G) = D4×C22⋊C4 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | D4×C22⋊C4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#577 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
668 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)(3,11)(4,16,18,8)(5,19)(6,10)(7,14)(9,20)(12,26,29,25)(13,23)(15,22)(17,24,32,31)(21,28,30,27);(1,3,6,15)(2,7,10,23)(4,12,18,29)(5,13,20,14)(8,21,16,30)(9,22,19,11)(17,27,32,28)(24,26,31,25);(1,4)(2,8)(3,12)(5,17)(6,18)(7,21)(9,24)(10,16)(11,25)(13,27)(14,28)(15,29)(19,31)(20,32)(22,26)(23,30);(3,13)(5,20)(7,22)(9,19)(11,23)(12,27)(14,15)(17,32)(21,26)(24,31)(25,30)(28,29)Polycyclic presentation:
2^8|a^2=g,b^2=h,e^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,d^b=de,e^b=eh,g^c=gh,e^d=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | #497324, #497326, #497330, … |
| maximal normal subgroup of | #497324, #497326, #497330, … |
| maximal quotient of | #220181, #220183, #220187, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15444);