Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61629 |
| 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 ◁ #525 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1029 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,16)(12,23)(13,18)(14,27)(15,19)(17,24)(20,35)(21,30)(22,31)(25,33)(26,32)(28,34)(29,38)(36,39)(37,40);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,20)(13,26)(14,21)(15,22)(17,29)(18,32)(19,31)(23,35)(24,38)(25,37)(27,30)(28,36)(33,40)(34,39);(9,12,15,25)(10,17,19,34)(11,20,22,37)(13,24,14,28)(16,29,31,39)(18,33,27,23)(21,36,26,38)(30,35,32,40);(3,8)(5,7)(9,13,15,14)(10,18,19,27)(11,21,22,26)(12,24,25,28)(16,30,31,32)(17,33,34,23)(20,36,37,38)(29,35,39,40);(2,6)(5,7)(11,22)(16,31)(20,37)(21,26)(29,39)(30,32)(35,40)(36,38)Polycyclic presentation:
2^8|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=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #152135, C2×#61637, #10577245, … |
| maximal normal subgroup of | #152135, C2×#61637, #10577245, … |
| maximal quotient of | #6386044, #6387646, #6387665, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54625);