Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of B(4,𝔽2) |
| automorphism | Aut(G) order 98304 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #120 |
| commutator | C23 | C2×C42 |
| Frattini, prime radical, cosocle | C25 | C23 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
629 conjugacy classes of subgroups; only the 151 normal subgroups are shown.
This group has 629 conjugacy classes of subgroups — too many to draw. Only its 151 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,4)(9,10)(11,17,19,14)(12,15)(13,16)(18,21,24,23)(20,22);(5,6,7,8)(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23);(1,2,3,4)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=efh,b^a=bd,c^a=ce,d^a=dh,e^b=eh,f^b=fh,d^c=dh64 conjugacy classes — view the full character table.
| Fitting subgroup of | #10936⋊C3 |
| maximal subgroup of | #130099, #130101, #130103, … |
| maximal normal subgroup of | #130099, #130101, #130103, … |
| maximal quotient of | #110045, #110047, #110402, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3924);