Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #15257 |
| automorphism | Aut(G) order 262144 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C24 |
| prime radical | C23×D4 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
881 conjugacy classes of subgroups; only the 167 normal subgroups are shown.
This group has 881 conjugacy classes of subgroups — too many to draw. Only its 167 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)(7,11)(10,12)(14,17)(15,19)(21,22)(23,27)(24,25)(26,28);(6,9)(10,12)(14,17)(18,20)(22,25)(23,28);(5,6)(7,10)(8,9)(11,12)(13,14)(15,18)(16,17)(19,20)(21,22)(23,26)(24,25)(27,28);(1,2)(3,4)(5,7)(6,10)(8,11)(9,12)(13,15,16,19)(14,18,17,20)(21,23,24,28)(22,26,25,27)Polycyclic presentation:
2^8|a^2=gh,d^2=ef,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de58 conjugacy classes — view the full character table.
| maximal subgroup of | #144421, C2×#16014, #609967, … |
| maximal normal subgroup of | #144421, C2×#16014, #609967, … |
| maximal quotient of | #154704, #154821, #155074, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 9002);