Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29592 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
388 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,3,4)(5,6,8,9)(7,10,11,12)(14,18)(15,21)(20,25)(23,26);(6,9)(10,12)(13,14,16,18)(15,22,21,28)(17,25,24,20)(19,26,27,23);(2,4)(7,11)(10,12)(13,15)(14,19)(16,21)(17,26)(18,27)(20,28)(22,25)(23,24);(5,7,8,11)(6,10,9,12)(15,21)(19,27)(22,28)(23,26)Polycyclic presentation:
2^8|a^2=ef,b^2=h,d^2=eh,g^2=h,b^a=be,c^a=cf,c^b=cg,g^b=gh,d^c=de,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#29833, #6435475, #6435486, … |
| maximal normal subgroup of | C2×#29833, #6435475, #6435486, … |
| maximal quotient of | #330415, #330416, #330417, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 22821);