Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #42996 |
| automorphism | Aut(G) order 65536 |
| 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 | |
785 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,4)(3,7,8,6)(9,10,14,13)(11,16,12,15)(17,18,21,22)(19,23,25,26)(20,28,29,24)(27,32,31,30);(9,11,14,12)(10,15,13,16)(18,22)(19,25)(20,29)(30,32);(1,3,5,8)(2,6,4,7)(9,12,14,11)(10,15,13,16)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31);(1,4,5,2)(3,7,8,6)(9,13,14,10)(11,15,12,16)(17,19)(18,23)(20,27)(21,25)(22,26)(24,30)(28,32)(29,31);(17,20,21,29)(18,24,22,28)(19,27,25,31)(23,30,26,32)Polycyclic presentation:
2^8|a^2=fgh,b^2=fh,c^2=fg,d^2=fg,e^2=h,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,d^c=dh58 conjugacy classes — view the full character table.
| maximal subgroup of | #7011395, #7011407, #7011454, … |
| maximal normal subgroup of | #7011395, #7011407, #7011454, … |
| maximal quotient of | #535504, #535551, #536193, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 38336);