Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61609 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #532 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1034 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(9,10,14,13)(11,18,17,15)(12,19,21,24)(16,23,20,22);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15)(12,19)(13,18)(14,17)(16,23)(20,22)(21,24);(9,12,14,21)(10,16,13,20)(11,19,17,24)(15,23,18,22);(3,8)(5,7)(9,13,14,10)(11,18,17,15)(12,20,21,16)(19,22,24,23);(2,6)(5,7)(10,13)(11,17)(16,20)(19,24)Polycyclic presentation:
2^8|a^2=h,c^2=h,d^2=h,g^2=h,b^a=bf,c^a=cg,e^a=ef,g^a=gh,d^b=dfh,e^b=eh,g^c=gh,e^d=eh49 conjugacy classes — view the full character table.
| maximal subgroup of | #244585, #244589, #244636, … |
| maximal normal subgroup of | #244585, #244589, #244636, … |
| maximal quotient of | #6386066, #6387747, #6388017, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54675);