Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | Aut(G) order 1179648 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C24 |
| commutator, Frattini, socle, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1072 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,8)(3,6,4,7)(10,13)(11,15)(17,18,21,22)(19,23,20,24);(1,2)(3,6)(4,7)(5,8)(9,10,12,13)(11,14,15,16)(18,22)(19,20);(1,3,5,4)(2,6,8,7)(9,11,12,15)(10,14,13,16)(17,19,21,20)(18,23,22,24);(1,3,5,4)(2,7,8,6)(10,13)(14,16)(17,20,21,19)(18,23,22,24);(1,4,5,3)(2,7,8,6)Polycyclic presentation:
2^8|a^2=fg,b^2=g,c^2=f,d^2=fg,e^2=fgh,b^a=bf,c^a=cg,d^a=dfg,c^b=ch,d^b=df64 conjugacy classes — view the full character table.
| Fitting subgroup of | #11710115, #11710116, #11710117 |
| maximal subgroup of | #97533, #129625, #131144, … |
| maximal normal subgroup of | #97533, #129625, #131144, … |
| maximal quotient of | #520968, #521166, C4×#2733, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 45295);