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 | #3489 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1058 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)(9,10,14,19)(11,15,22,29)(12,17,23,31)(13,16,25,18)(20,27,33,38)(21,26,35,28)(24,30,36,32)(34,37,40,39);(2,5)(6,8)(9,11,14,22)(10,15,19,29)(12,20,23,33)(13,21,25,35)(16,26,18,28)(17,27,31,38)(24,34,36,40)(30,37,32,39);(1,3,4,7)(2,6,5,8)(10,16)(13,25)(15,26)(17,30)(18,19)(21,35)(24,36)(27,37)(28,29)(31,32)(34,40)(38,39);(9,12)(10,17)(11,20)(13,24)(14,23)(15,27)(16,30)(18,32)(19,31)(21,34)(22,33)(25,36)(26,37)(28,39)(29,38)(35,40);(12,23)(17,31)(20,33)(24,36)(27,38)(30,32)(34,40)(37,39)Polycyclic presentation:
2^8|a^2=h,b^2=h,c^2=f,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 | #60524⋊C3 |
| maximal subgroup of | #6574957, #6574960, #6574961, … |
| maximal normal subgroup of | #6574957, #6574960, #6574961, … |
| maximal quotient of | #6370912, #6371346, #6371391, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53512);