Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29592 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
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 | |
532 conjugacy classes of subgroups; only the 131 normal subgroups are shown.
This group has 532 conjugacy classes of subgroups — too many to draw. Only its 131 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:
(5,6,8,12)(7,11,9,10)(13,14,17,20)(15,21,23,18)(16,19,24,26)(22,27,28,25);(1,2)(3,4)(5,7,8,9)(6,10,12,11)(13,15,17,23)(14,18,20,21)(16,22,24,28)(19,25,26,27);(2,4)(6,10)(7,9)(11,12)(13,16)(14,19)(15,22)(17,24)(18,25)(20,26)(21,27)(23,28);(5,7,8,9)(6,11,12,10)(16,24)(19,26)(22,28)(25,27)Polycyclic presentation:
2^8|a^2=e,b^2=e,d^2=h,f^2=h,b^a=be,c^a=cf,f^a=fh,c^b=cg,d^c=de,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #166746, #166752, #166778, … |
| maximal normal subgroup of | #166746, #166752, #166778, … |
| maximal quotient of | #327777, #327785, #327793, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 22632);