Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| 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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1093 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6,10,12)(7,9,8,11)(13,14,18,21)(15,16,19,23)(17,20,25,27)(22,24,26,28);(5,7,10,8)(6,11,12,9)(15,19)(16,23)(17,25)(20,27);(1,2,3,4)(6,12)(7,8)(13,15)(14,16)(17,22)(18,19)(20,24)(21,23)(25,26)(27,28);(5,8,10,7)(6,11,12,9)(13,16)(14,19)(15,21)(17,24)(18,23)(20,26)(22,27)(25,28);(5,9,10,11)(6,7,12,8)(13,17)(14,20)(15,22)(16,24)(18,25)(19,26)(21,27)(23,28)Polycyclic presentation:
2^8|a^2=h,b^2=f,c^2=fg,d^2=f,e^2=f,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dfh,e^b=eh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#48835, #7704928, #7704955, … |
| maximal normal subgroup of | C2×#48835, #7704928, #7704955, … |
| maximal quotient of | #560419, #560425, #561525, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 41823);