Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60768 |
| 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 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
884 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,9)(4,8)(5,7)(6,10)(11,15)(12,14)(13,16)(17,18)(19,26)(20,28)(21,25)(22,23)(24,30)(27,31)(29,32);(1,3)(2,6)(4,11)(5,10)(7,9)(8,13)(12,16)(14,15)(17,19,21,22)(18,23,25,26)(20,29,30,27)(24,32,28,31);(3,10)(6,9)(11,16)(13,15)(18,23)(19,22)(24,31)(25,26)(27,29)(28,32);(2,7)(6,9)(8,14)(13,15)(17,20,21,30)(18,24,25,28)(19,27,22,29)(23,31,26,32);(1,4,5,12)(2,8,7,14)(3,11,10,16)(6,13,9,15)(20,30)(24,28)(27,29)(31,32)Polycyclic presentation:
2^8|b^2=h,d^2=h,e^2=fh,g^2=h,b^a=bf,c^a=cg,d^a=df,g^a=gh,c^b=cf,d^b=dh,g^c=gh,e^d=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | #131643, #131645, #485031, … |
| maximal normal subgroup of | #131643, #131645, #485031, … |
| maximal quotient of | #6374693, #6375044, #6376032, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54002);