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 | |
1065 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(13,14)(15,17)(16,20)(18,26)(19,22)(21,27)(23,28)(24,25);(5,6)(7,11)(8,12)(9,10)(13,15,19,16)(14,20,22,17)(18,23,27,24)(21,28,26,25);(1,2)(3,4)(6,10)(7,12)(13,16,19,15)(14,20,22,17)(18,24,27,23)(21,28,26,25);(5,7)(6,8)(9,12)(10,11)(13,17,19,20)(14,15,22,16)(18,25,27,28)(21,23,26,24);(5,8)(6,7)(9,11)(10,12)(13,18)(14,21)(15,23)(16,24)(17,25)(19,27)(20,28)(22,26)Polycyclic presentation:
2^8|a^2=g,b^2=f,c^2=f,d^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×#47859, #7703570, #7704091, … |
| maximal normal subgroup of | C2×#47859, #7703570, #7704091, … |
| maximal quotient of | #560505, #560549, #561603, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 40847);