Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23903 |
| automorphism | Aut(G) = C22×#72217875 (order 4096) · Inn(G) = C2×#120 · Out(G) = C23×D4 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C24 | C24 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
646 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 646 conjugacy classes of subgroups — too many to draw. Only its 123 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) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,7)(4,5)(6,8)(9,10)(11,14)(12,15)(13,16)(17,22)(18,21)(19,23)(20,24);(2,5)(6,7)(9,11)(10,14)(12,17)(13,18)(15,22)(16,21)(19,24)(20,23);(9,12)(10,15)(11,17)(13,19)(14,20)(16,23)(18,24)(21,22);(1,3,4,8)(2,6,5,7)(12,19)(15,23)(17,24)(20,22)Polycyclic presentation:
2^8|d^2=e,b^a=be,c^a=cf,d^a=de,g^a=gh,c^b=cg,f^b=fh,d^c=dh40 conjugacy classes — view the full character table.
| maximal subgroup of | #245521, #245525, #245533, … |
| maximal normal subgroup of | #245521, #245525, #245533, … |
| maximal quotient of | #258051, #258052, #258055, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17130);