Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2812 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C2×C43 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C24 |
| commutator | C23 | C23×C4 |
| prime radical | C25 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
645 conjugacy classes of subgroups; only the 187 normal subgroups are shown.
This group has 645 conjugacy classes of subgroups — too many to draw. Only its 187 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,3,4)(14,17)(18,20)(22,25)(26,27);(6,9)(10,12)(14,17)(15,19)(21,22,24,25)(23,26,28,27);(5,6)(7,10)(8,9)(11,12)(13,14,16,17)(15,18,19,20)(21,22)(23,27)(24,25)(26,28);(5,7,8,11)(6,10,9,12)(13,15,16,19)(14,18,17,20)(21,23,24,28)(22,26,25,27)Polycyclic presentation:
2^8|a^2=h,b^2=e,c^2=ef,d^2=g,b^a=be,c^a=cf,c^b=cg,d^b=def,d^c=de52 conjugacy classes — view the full character table.
| maximal subgroup of | #140051, C2×#14649, #544300, … |
| maximal normal subgroup of | #140051, C2×#14649, #544300, … |
| maximal quotient of | #154059, #154061, #154062, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 7637);