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 | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
932 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)(5,6,10,8)(7,11,13,15)(9,12,17,16)(14,18,19,20)(21,22)(23,26)(24,32)(25,28)(27,33)(29,31)(30,35)(34,36);(7,13)(9,17)(11,15)(12,16)(21,22,25,28)(23,29,31,26)(24,32,33,27)(30,34,36,35);(1,2,3,4)(5,7)(6,11)(8,15)(9,14)(10,13)(12,18)(16,20)(17,19)(21,22)(23,29)(24,27)(25,28)(26,31)(30,35)(32,33)(34,36);(5,8,10,6)(7,11,13,15)(9,16,17,12)(14,18,19,20)(21,23,25,31)(22,26,28,29)(24,30,33,36)(27,34,32,35);(5,9)(6,12)(7,14)(8,16)(10,17)(11,18)(13,19)(15,20)(21,24,25,33)(22,27,28,32)(23,30,31,36)(26,34,29,35)Polycyclic presentation:
2^8|a^2=fh,b^2=f,c^2=g,d^2=h,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×#47767, #7701759, #7701779, … |
| maximal normal subgroup of | C2×#47767, #7701759, #7701779, … |
| maximal quotient of | #576645, #576694, #576709, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 40755);