Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35346 |
| automorphism | Aut(G) order 524288 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C2×#537 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1171 conjugacy classes of subgroups; only the 487 normal subgroups are shown.
This group has 1171 conjugacy classes of subgroups — too many to draw. Only its 487 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 — 5 generators on 28 points:
(2,4)(5,6,9,7)(8,11,12,10)(13,14)(15,18)(16,19)(17,20)(21,24)(22,25)(23,26)(27,28);(1,2,3,4)(5,7,9,6)(8,11,12,10)(14,15)(17,25)(19,21)(23,28);(5,8,9,12)(6,10,7,11)(13,15,18,14)(16,19,24,21)(17,22,25,20)(23,26,28,27);(13,16,18,24)(14,19,15,21)(17,23,25,28)(20,26,22,27);(13,17)(14,20)(15,22)(16,23)(18,25)(19,26)(21,27)(24,28)Polycyclic presentation:
2^8|a^2=gh,b^2=fg,c^2=g,d^2=h,b^a=bf,c^a=cg,c^b=ch,e^b=eh,d^c=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #97485, #132160, #133764, … |
| maximal normal subgroup of | #97485, #132160, #133764, … |
| maximal quotient of | #515340, #515397, #515539, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 28637);