Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #46350 |
| 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 | #2682 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1028 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)(7,11)(8,9)(10,12)(13,14,18,21)(15,16,19,24)(17,20,26,28)(22,23,25,27);(5,7,9,12)(6,10,8,11)(13,15,18,19)(14,16,21,24)(17,25,26,22)(20,27,28,23);(1,2,3,4)(7,12)(10,11)(13,15)(14,16)(17,22)(18,19)(20,23)(21,24)(25,26)(27,28);(5,7)(6,11)(8,10)(9,12)(13,16)(14,19)(15,21)(17,23)(18,24)(20,25)(22,28)(26,27);(5,8,9,6)(7,10,12,11)(13,17)(14,20)(15,22)(16,23)(18,26)(19,25)(21,28)(24,27)Polycyclic presentation:
2^8|a^2=fh,b^2=h,c^2=g,e^2=f,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=df,e^b=eh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#47208, #7700965, #7700968, … |
| maximal normal subgroup of | C2×#47208, #7700965, #7700968, … |
| maximal quotient of | #576618, #576670, #576733, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 40196);