Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | Aut(G) order 65536 |
| 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 | |
1014 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)(7,10)(8,13)(9,12)(11,16)(14,17)(15,19)(18,20)(21,22,26,25)(23,29,31,27)(24,32,34,28)(30,35,33,36);(6,7)(11,13)(12,14)(18,19)(22,25)(27,29)(28,32)(35,36);(1,2)(3,4)(5,7,10,6)(8,13,16,11)(9,14,17,12)(15,19,20,18)(21,23,26,31)(22,27,25,29)(24,33,34,30)(28,36,32,35);(5,8,10,16)(6,11,7,13)(9,15,17,20)(12,18,14,19)(21,24,26,34)(22,28,25,32)(23,30,31,33)(27,35,29,36);(5,9,10,17)(6,12,7,14)(8,15,16,20)(11,18,13,19)(21,25,26,22)(23,29,31,27)(24,32,34,28)(30,36,33,35)Polycyclic presentation:
2^8|a^2=fh,c^2=f,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,d^c=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | #215019, #6352416, #6352531, … |
| maximal normal subgroup of | #215019, #6352416, #6352531, … |
| maximal quotient of | #544493, #545057, #545418, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 34373);