Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61049 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
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 | |
1357 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2)(3,4)(5,6,11,16)(7,17,21,34)(8,13,24,32)(9,15,10,14)(12,26,29,20)(18,28,22,36)(19,27,35,30)(23,33,25,31);(5,7,11,21)(6,12,16,29)(8,22,24,18)(9,20,10,26)(13,30,32,27)(14,17,15,34)(19,25,35,23)(28,33,36,31);(2,4)(5,8)(6,13)(7,18)(9,25)(10,23)(11,24)(12,27)(14,33)(15,31)(16,32)(17,28)(19,20)(21,22)(26,35)(29,30)(34,36);(5,8)(6,13)(7,19)(9,23)(10,25)(11,24)(12,28)(14,31)(15,33)(16,32)(17,27)(18,20)(21,35)(22,26)(29,36)(30,34);(5,9,11,10)(6,14,16,15)(7,20,21,26)(8,23,24,25)(12,17,29,34)(13,31,32,33)(18,35,22,19)(27,36,30,28)Polycyclic presentation:
2^8|a^2=h,b^2=h,e^2=h,f^2=h,b^a=bf,c^a=cg,e^a=eh,f^a=fh,c^b=ch,d^b=df,f^b=fh,e^c=eh,f^d=fh58 conjugacy classes — view the full character table.
| maximal subgroup of | #130787, #130789, #130790, … |
| maximal normal subgroup of | #130787, #130789, #130790, … |
| maximal quotient of | #6359118, #6359126, #6359127, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54078);