Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60372 |
| 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 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1339 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6)(7,15)(8,13)(9,11)(10,22)(12,18)(14,21)(16,29)(17,28)(19,25)(20,33)(23,24)(26,31)(27,32)(30,36)(34,35);(1,2,3,4)(6,11)(12,24)(13,25)(14,22)(15,28)(26,35)(27,33)(29,36);(5,7,9,17)(6,12,11,24)(8,16,19,30)(10,23,21,18)(13,26,25,35)(14,28,22,15)(20,34,32,31)(27,36,33,29);(5,8)(6,13)(7,16)(9,19)(10,20)(11,25)(12,26)(14,27)(15,29)(17,30)(18,31)(21,32)(22,33)(23,34)(24,35)(28,36);(8,19)(13,25)(16,30)(20,32)(26,35)(27,33)(29,36)(31,34)Polycyclic presentation:
2^8|b^2=fh,c^2=h,g^2=h,b^a=bf,c^a=cg,g^a=gh,g^c=gh,e^d=eh61 conjugacy classes — view the full character table.
| Fitting subgroup of | #60433⋊C3 |
| maximal subgroup of | #6573985, #6573988, #6573990, … |
| maximal normal subgroup of | #6573985, #6573988, #6573990, … |
| maximal quotient of | #6356029, #6356030, #6356031, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53421);