Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23924 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
394 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,8)(3,7,4,6)(9,10,12,14)(11,13,15,16)(17,18,21,25)(19,22,26,30)(20,23,28,24)(27,29,32,31);(1,3,5,4)(2,6,8,7)(9,11)(10,13)(12,15)(14,16)(17,19)(18,22)(20,27)(21,26)(23,29)(24,31)(25,30)(28,32);(11,15)(13,16)(18,23)(20,28)(22,29)(24,25)(27,32)(30,31);(1,4,5,3)(2,7,8,6)(19,26)(22,30)(27,32)(29,31)Polycyclic presentation:
2^8|a^2=egh,b^2=e,d^2=e,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 | C2×#27688, #6458237, #6458244, … |
| maximal normal subgroup of | C2×#27688, #6458237, #6458244, … |
| maximal quotient of | #314998, #315002, #315006, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 20676);