Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| automorphism | Aut(G) order 393216 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C24×C43 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1128 conjugacy classes of subgroups; only the 407 normal subgroups are shown.
This group has 1128 conjugacy classes of subgroups — too many to draw. Only its 407 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 — 5 generators on 24 points:
(1,2,4,5)(3,7,8,6)(9,10)(11,15)(12,13)(14,16)(18,20)(19,23);(9,10,12,13)(11,14,16,15)(17,18,21,20)(19,24,23,22);(2,5)(6,7)(10,13)(14,15)(19,23)(22,24);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,22)(20,24)(21,23);(1,3,4,8)(2,6,5,7)(9,11,12,16)(10,14,13,15)(17,20,21,18)(19,24,23,22)Polycyclic presentation:
2^8|a^2=gh,b^2=f,e^2=fgh,b^a=bf,c^a=cg,d^a=dfg,e^a=efgh,c^b=ch,d^b=df,d^c=dfgh46 conjugacy classes — view the full character table.
| Fitting subgroup of | #53015⋊C3 |
| maximal subgroup of | #132654, #132665, #487391, … |
| maximal normal subgroup of | #132654, #132665, #487391, … |
| maximal quotient of | #582148, #582364, #582576, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 46003);