Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #34980 |
| automorphism | Aut(G) order 262144 |
| 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 | #2619 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1101 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)(6,9)(10,12)(14,18)(16,23)(19,24)(22,27);(5,6)(7,10)(8,9)(11,12)(14,18)(15,21)(20,25)(22,27);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,19)(16,20)(17,18)(21,24)(22,26)(23,25)(27,28);(5,7,8,11)(6,10,9,12)(13,15)(14,19)(16,22)(17,21)(18,24)(20,26)(23,27)(25,28);(13,16,17,23)(14,20,18,25)(15,22,21,27)(19,26,24,28)Polycyclic presentation:
2^8|c^2=fgh,d^2=f,e^2=h,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #129917, #208580, #208586, … |
| maximal normal subgroup of | #129917, #208580, #208586, … |
| maximal quotient of | #516449, #516627, #516629, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 28319);