Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18376 |
| automorphism | Aut(G) = C22×#72217875 (order 4096) · Inn(G) = C2×#120 · Out(G) = C23×D4 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #3505 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
506 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 506 conjugacy classes of subgroups — too many to draw. Only its 119 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:
(1,2,7,4)(3,13,18,28)(5,11,6,20)(8,12,24,21)(9,15,14,16)(10,22,19,23)(17,27,32,25)(26,31,30,29);(1,3)(2,9)(4,14)(5,16)(6,15)(7,18)(8,17)(10,27)(11,28)(12,26)(13,20)(19,25)(21,30)(22,29)(23,31)(24,32);(1,4,7,2)(3,14,18,9)(5,19,6,10)(8,21,24,12)(11,23,20,22)(13,29,28,31)(15,27,16,25)(17,30,32,26);(1,5,7,6)(2,10,4,19)(3,15,18,16)(8,22,24,23)(9,25,14,27)(11,12,20,21)(13,26,28,30)(17,31,32,29)Polycyclic presentation:
2^8|a^2=g,c^2=g,d^2=g,e^2=g,b^a=be,c^a=cf,d^a=dgh,e^a=eg,d^b=dg,e^b=eg,f^b=fh,e^c=eh43 conjugacy classes — view the full character table.
| maximal subgroup of | #245524, #245525, #245536, … |
| maximal normal subgroup of | #245524, #245525, #245536, … |
| maximal quotient of | #169622, #169624, #169634, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11373);