Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33388 |
| automorphism | Aut(G) order 24576 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C22×#102 |
| commutator | C8 | C23×C4 |
| Frattini, cosocle | C2×C8 | C24 |
| prime radical | C2×#576 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×#41 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
577 conjugacy classes of subgroups; only the 221 normal subgroups are shown.
This group has 577 conjugacy classes of subgroups — too many to draw. Only its 221 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,10,6,12,20,11)(3,8,15,25,13,23,28,26)(4,16,18,9,19,7,17,22)(14,29,31,24,27,21,30,32);(2,7)(4,17)(5,20)(8,21)(9,10)(11,22)(12,16)(14,30)(15,28)(18,19)(23,29)(24,25)(26,32)(27,31);(1,3)(2,8)(4,14)(5,15)(6,13)(7,21)(9,24)(10,25)(11,26)(12,23)(16,29)(17,30)(18,31)(19,27)(20,28)(22,32);(3,13)(8,23)(14,27)(15,28)(21,29)(24,32)(25,26)(30,31)Polycyclic presentation:
2^8|a^2=f,e^2=g,f^2=h,g^2=h,b^a=be,e^a=eg,g^a=gh,e^b=egh,f^b=fh,g^b=gh,d^c=dh58 conjugacy classes — view the full character table.
| Fitting subgroup of | #33392⋊C3 |
| maximal subgroup of | #497410, #497411, #498124, … |
| maximal normal subgroup of | #497410, #497411, #498124, … |
| maximal quotient of | #481066, #481284, #481319, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26380);