Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30412 |
| 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 | #2431 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
500 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)(7,11)(8,10)(9,12)(14,19)(20,27)(21,22)(23,28);(5,7)(6,9)(8,12)(10,11)(13,14,17,19)(15,23,25,28)(16,22,18,21)(20,26,27,24);(2,4)(13,15)(14,20)(16,24)(17,25)(18,26)(19,27)(21,23)(22,28);(2,4)(6,10)(7,12)(13,16,17,18)(14,21,19,22)(15,24,25,26)(20,23,27,28)Polycyclic presentation:
2^8|a^2=f,b^2=h,d^2=h,g^2=h,b^a=be,c^a=cf,d^a=defh,c^b=cg,d^b=de,g^b=gh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #6632160, #6632167, #6632177, … |
| maximal normal subgroup of | #6632160, #6632167, #6632177, … |
| maximal quotient of | #342525, #342526, #342529, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23993);