Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2812 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C24 |
| commutator, socle | C23 | C23×C4 |
| prime radical | C22×D4 | C2×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
343 conjugacy classes of subgroups; only the 145 normal subgroups are shown.
This group has 343 conjugacy classes of subgroups — too many to draw. Only its 145 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,5)(4,6)(7,8)(10,13)(11,15)(17,18,21,24,20,23,28,31)(19,25,27,32,26,22,29,30);(1,3)(2,5)(4,7)(6,8)(9,10)(11,14)(12,13)(15,16);(2,6)(3,7)(9,11,12,15)(10,14,13,16)(17,19)(18,22)(20,26)(21,27)(23,25)(24,30)(28,29)(31,32);(2,6)(5,8)(10,13)(14,16)(19,26)(22,25)(27,29)(30,32)Polycyclic presentation:
2^8|a^2=g,c^2=e,g^2=h,b^a=be,c^a=cf,d^a=defh,c^b=cefh,d^b=de,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #136291, #137284, #139957, … |
| maximal normal subgroup of | #136291, #137284, #139957, … |
| maximal quotient of | #357292, #357325, #357346, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16311);