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 |
| prime radical | #392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
963 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6,10,11)(7,13,15,8)(9,17,18,12)(14,16,20,19)(22,26)(24,32)(27,29)(30,36);(6,11)(8,13)(12,17)(16,19)(21,22,25,26)(23,27,31,29)(24,33,32,28)(30,35,36,34);(1,2,3,4)(5,7)(6,8)(9,14)(10,15)(11,13)(12,16)(17,19)(18,20)(21,22)(23,29)(24,28)(25,26)(27,31)(30,35)(32,33)(34,36);(5,8)(6,7)(9,16)(10,13)(11,15)(12,14)(17,20)(18,19)(21,23,25,31)(22,27,26,29)(24,30,32,36)(28,34,33,35);(5,9,10,18)(6,12,11,17)(7,14,15,20)(8,16,13,19)(21,24,25,32)(22,28,26,33)(23,30,31,36)(27,34,29,35)Polycyclic presentation:
2^8|a^2=fh,b^2=h,c^2=fg,d^2=h,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×#51205, #7706880, #7706886, … |
| maximal normal subgroup of | C2×#51205, #7706880, #7706886, … |
| maximal quotient of | #560446, #560476, #561541, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 44193);