Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35346 |
| 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 | #2543 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1100 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)(6,10)(7,12)(13,14,18,19)(15,20,22,27)(16,17,21,26)(23,28,25,24);(5,6)(7,11)(8,12)(9,10)(14,19)(15,22)(17,26)(23,25);(1,2,3,4)(5,7,9,12)(6,11,10,8)(13,15)(14,20)(16,25)(17,24)(18,22)(19,27)(21,23)(26,28);(5,8,9,11)(6,12,10,7)(13,16,18,21)(14,17,19,26)(15,23,22,25)(20,28,27,24);(13,17)(14,21)(15,24)(16,19)(18,26)(20,23)(22,28)(25,27)Polycyclic presentation:
2^8|a^2=h,c^2=g,d^2=f,b^a=bf,c^a=cg,c^b=ch,e^b=eh,d^c=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#35464, #7628839, #7628865, … |
| maximal normal subgroup of | C2×#35464, #7628839, #7628865, … |
| maximal quotient of | #515344, #515357, #515474, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 28452);