Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60372 |
| 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 |
| prime radical | #3010 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1007 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)(3,6)(4,5)(7,8)(9,10)(11,20)(12,23)(13,16)(14,18)(15,29)(17,26)(19,28)(21,37)(22,32)(24,40)(25,34)(27,36)(30,33)(31,39)(35,38);(2,5)(6,8)(9,11,14,13)(10,16,18,20)(12,21,25,24)(15,22,28,27)(17,31,33,35)(19,32,29,36)(23,40,34,37)(26,38,30,39);(9,12,14,25)(10,17,18,33)(11,21,13,24)(15,30,28,26)(16,31,20,35)(19,34,29,23)(22,39,27,38)(32,37,36,40);(1,3,4,7)(2,6,5,8)(9,13,14,11)(10,16,18,20)(12,24,25,21)(15,27,28,22)(17,31,33,35)(19,32,29,36)(23,40,34,37)(26,39,30,38);(11,13)(16,20)(21,24)(22,27)(31,35)(32,36)(37,40)(38,39)Polycyclic presentation:
2^8|b^2=h,c^2=h,d^2=f,g^2=h,b^a=bf,c^a=cg,g^a=gh,e^b=eh,g^c=gh,e^d=eh61 conjugacy classes — view the full character table.
| maximal subgroup of | #6576086, #6576093, #6576098, … |
| maximal normal subgroup of | #6576086, #6576093, #6576098, … |
| maximal quotient of | #6370993, #6371817, #6371818, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53583);