Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29730 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
508 conjugacy classes of subgroups; only the 139 normal subgroups are shown.
This group has 508 conjugacy classes of subgroups — too many to draw. Only its 139 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:
(5,6)(7,10)(8,9)(11,12)(13,14,16,20,17,21,18,22)(15,23,25,27,24,19,26,28);(1,2,3,4)(6,9)(10,12)(13,15)(14,19)(16,25)(17,24)(18,26)(20,28)(21,23)(22,27);(2,4)(5,7)(6,10)(8,11)(9,12)(14,20)(15,24)(16,18)(19,27)(21,22)(23,28);(7,11)(10,12)(13,16,17,18)(14,20,21,22)(15,25,24,26)(19,28,23,27)Polycyclic presentation:
2^8|a^2=fh,b^2=gh,d^2=h,f^2=h,b^a=be,c^a=cf,c^b=cg,d^c=de,f^c=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | #170141, #170147, #170173, … |
| maximal normal subgroup of | #170141, #170147, #170173, … |
| maximal quotient of | #331100, #331106, #331107, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23232);