Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #14715 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C24 |
| prime radical | C25 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
681 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,3)(6,9)(10,12)(13,14,17,15)(16,19,20,18)(21,22,25,24)(23,27,28,26);(1,2,4,3)(14,15)(18,19)(23,28)(26,27);(1,3,4,2)(5,6)(7,10)(8,9)(11,12)(13,15,17,14)(16,19,20,18)(21,23)(22,26)(24,27)(25,28);(5,7,8,11)(6,10,9,12)(13,16,17,20)(14,18,15,19)(21,24,25,22)(23,27,28,26)Polycyclic presentation:
2^8|a^2=g,b^2=eh,c^2=e,d^2=ef,b^a=be,c^a=cf,d^a=dh,c^b=cg64 conjugacy classes — view the full character table.
| maximal subgroup of | #296491, #296493, #296496, … |
| maximal normal subgroup of | #296491, #296493, #296496, … |
| maximal quotient of | #154148, #154212, #154279, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 8035);