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 | #2656 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1108 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6,8,10)(7,9,12,11)(13,14,18,19)(15,22,20,16)(17,21,25,26)(23,28,27,24);(2,4)(5,6,8,10)(7,11,12,9)(14,19)(15,20)(21,26)(23,27);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,15,18,20)(14,16,19,22)(17,23,25,27)(21,24,26,28);(13,16)(14,20)(15,19)(17,24)(18,22)(21,27)(23,26)(25,28);(1,2)(3,4)(5,7,8,12)(6,9,10,11)(13,17,18,25)(14,21,19,26)(15,23,20,27)(16,24,22,28)Polycyclic presentation:
2^8|a^2=g,b^2=fg,c^2=f,e^2=g,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 | #97784, #246035, #246036, … |
| maximal normal subgroup of | #97784, #246035, #246036, … |
| maximal quotient of | #521347, #522078, #522604, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 30759);