Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| 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 |
| prime radical | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1010 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(13,14)(15,23)(16,17)(18,20)(19,22)(21,28)(24,25)(26,27);(6,9)(7,11)(13,15,19,18)(14,20,22,23)(16,26,21,24)(17,25,28,27);(1,2)(3,4)(5,6,8,9)(7,10,11,12)(13,16,19,21)(14,17,22,28)(15,24,18,26)(20,27,23,25);(13,17)(14,21)(15,25)(16,22)(18,27)(19,28)(20,24)(23,26);(5,7)(6,10)(8,11)(9,12)(13,18,19,15)(14,20,22,23)(16,26,21,24)(17,27,28,25)Polycyclic presentation:
2^8|a^2=g,b^2=f,c^2=h,e^2=f,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh58 conjugacy classes — view the full character table.
| maximal subgroup of | #481005, #6542416, #6572630, … |
| maximal normal subgroup of | #481005, #6542416, #6572630, … |
| maximal quotient of | #521660, #522291, #522778, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 31033);