Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| automorphism | Aut(G) = C215 (order 32768) · Inn(G) = C25 · Out(G) = C210 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
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 | |
743 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,6)(3,7,8,4)(10,13)(14,16)(17,18,22,25)(19,20,26,23)(21,24,29,31)(27,28,32,30);(1,3,5,8)(2,4,6,7)(19,26)(20,23)(21,29)(24,31);(1,4)(2,3)(5,7)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,23)(20,25)(21,27)(22,26)(24,30)(28,31)(29,32);(1,2)(3,4)(5,6)(7,8)(17,20,22,23)(18,19,25,26)(21,28,29,30)(24,27,31,32);(1,3)(2,4)(5,8)(6,7)(9,11,12,15)(10,14,13,16)(17,21,22,29)(18,24,25,31)(19,27,26,32)(20,28,23,30)Polycyclic presentation:
2^8|a^2=h,b^2=f,d^2=fh,e^2=g,b^a=bf,c^a=cg,d^a=dh,e^a=ef,c^b=ch,d^b=dfh,e^b=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#46189, #7702771, #7702846, … |
| maximal normal subgroup of | C2×#46189, #7702771, #7702846, … |
| maximal quotient of | #557322, #557333, #557537, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 39177);