Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×D4 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
477 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,3)(5,6,8,7)(9,10)(11,17)(12,20)(13,15)(14,18)(16,22)(19,24)(21,23);(1,3,4,2)(6,7);(9,11,13,19)(10,14,15,23)(12,21,22,18)(16,24,20,17);(5,7,8,6)(10,15)(14,23)(16,20)(17,24)Polycyclic presentation:
2^8|a^2=g,b^2=eg,c^2=h,d^2=e,f^2=h,b^a=be,c^a=cf,d^a=dh,f^a=fh,d^b=de,f^c=fh58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#23009, #6381850, #6381852, … |
| maximal normal subgroup of | C2×#23009, #6381850, #6381852, … |
| maximal quotient of | #229624, #229632, #229656, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15997);