Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23996 |
| automorphism | Aut(G) = C2×#72216396 (order 2048) · Inn(G) = #2404 · Out(G) = C2×D4 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2404 |
| commutator, Frattini, cosocle | C24 | C24 |
| prime radical | #3026 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
605 conjugacy classes of subgroups; only the 121 normal subgroups are shown.
This group has 605 conjugacy classes of subgroups — too many to draw. Only its 121 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)(3,7)(4,8)(5,9)(6,10)(11,18)(12,19)(13,20)(14,26)(15,28)(16,22)(17,23)(21,29)(24,32)(25,31)(27,30);(1,3,6,13)(2,7,10,20)(4,14)(5,12,17,25)(8,18)(9,19,23,31)(11,16)(15,27)(21,30)(22,26)(24,29)(28,32);(1,4)(2,8)(3,11)(5,15)(6,16)(7,18)(9,21)(10,22)(12,24)(13,14)(17,29)(19,30)(20,26)(23,28)(25,27)(31,32);(1,5)(2,9)(3,12)(4,15)(6,17)(7,19)(8,21)(10,23)(11,24)(13,25)(14,27)(16,29)(18,30)(20,31)(22,28)(26,32)Polycyclic presentation:
2^8|b^2=eh,b^a=be,c^a=cf,d^a=de,g^a=gh,c^b=cg,g^b=gh,e^c=eh,f^d=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #511854, #511855, #511857, … |
| maximal normal subgroup of | #511854, #511855, #511857, … |
| maximal quotient of | #251832, #251833, #251834, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16985);