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 | C22×#41 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1131 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)(5,6,9,12)(7,8,11,10);(5,7,9,11)(6,10,12,8)(14,18)(19,25)(20,26)(27,28);(1,2)(3,4)(5,6)(7,10)(8,11)(9,12)(13,14)(15,19)(16,20)(17,18)(21,27)(22,25)(23,28)(24,26);(5,8,9,10)(6,11,12,7)(13,15,17,22)(14,19,18,25)(16,23,24,21)(20,28,26,27);(13,16,17,24)(14,20,18,26)(15,21,22,23)(19,27,25,28)Polycyclic presentation:
2^8|a^2=f,b^2=f,d^2=fh,e^2=h,b^a=bf,c^a=cg,c^b=ch,e^d=eh64 conjugacy classes — view the full character table.
| maximal subgroup of | #129907, #214795, #214843, … |
| maximal normal subgroup of | #129907, #214795, #214843, … |
| maximal quotient of | #517640, #517642, #517671, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 28700);