Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35680 |
| automorphism | Aut(G) order 131072 |
| 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 | #2540 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1076 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,3)(5,6,9,12)(7,10,8,11)(13,14)(15,20)(16,21)(17,22)(18,19)(23,25)(24,28)(26,27);(1,2,4,3)(5,7,9,8)(6,10,12,11)(14,19)(15,21)(22,27)(23,28);(1,3,4,2)(5,6)(7,10)(8,11)(9,12)(13,15,18,21)(14,20,19,16)(17,23,26,28)(22,25,27,24);(5,7,9,8)(6,10,12,11)(13,16,18,20)(14,21,19,15)(17,25,26,24)(22,23,27,28);(5,8,9,7)(6,11,12,10)(13,17,18,26)(14,22,19,27)(15,23,21,28)(16,24,20,25)Polycyclic presentation:
2^8|a^2=fgh,b^2=g,c^2=f,d^2=fg,e^2=fg,b^a=bf,c^a=cg,c^b=ch,e^d=eh64 conjugacy classes — view the full character table.
| maximal subgroup of | #132401, #248320, #248321, … |
| maximal normal subgroup of | #132401, #248320, #248321, … |
| maximal quotient of | #517894, #517897, #517905, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 29036);