Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | Aut(G) = C215 (order 32768) · Inn(G) = C25 · Out(G) = C210 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | #392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1012 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(13,14,18,19)(15,16,21,24)(17,20,25,26)(22,23,27,28);(6,10)(8,11)(14,19)(15,21)(20,26)(22,27);(1,2,3,4)(5,6)(7,11)(8,12)(9,10)(13,15)(14,16)(17,22)(18,21)(19,24)(20,23)(25,27)(26,28);(5,7,9,12)(6,11,10,8)(13,16)(14,15)(17,23)(18,24)(19,21)(20,22)(25,28)(26,27);(5,8,9,11)(6,12,10,7)(13,17,18,25)(14,20,19,26)(15,22,21,27)(16,23,24,28)Polycyclic presentation:
2^8|a^2=f,c^2=g,d^2=fh,e^2=h,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#39945, #7637650, #7638119, … |
| maximal normal subgroup of | C2×#39945, #7637650, #7638119, … |
| maximal quotient of | #521965, #522504, #523057, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 32933);