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) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
561 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(5,6,9,7)(8,10,12,11)(14,17)(18,20)(21,22,24,26)(23,25,27,28);(6,7)(10,11)(13,14)(15,18)(16,17)(19,20)(21,23,24,27)(22,25,26,28);(1,2,3,4)(5,7,9,6)(8,11,12,10)(13,14)(15,18)(16,17)(19,20)(21,23)(22,25)(24,27)(26,28);(5,8,9,12)(6,10,7,11)(13,15,16,19)(14,18,17,20)(21,22)(23,25)(24,26)(27,28)Polycyclic presentation:
2^8|a^2=g,b^2=h,c^2=ef,d^2=e,b^a=be,c^a=cf,d^a=dh,c^b=cg64 conjugacy classes — view the full character table.
| maximal subgroup of | #294493, #294494, #294513, … |
| maximal normal subgroup of | #294493, #294494, #294513, … |
| maximal quotient of | #154138, #154234, #154279, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 8012);