Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| 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 | |
363 conjugacy classes of subgroups; only the 153 normal subgroups are shown.
This group has 363 conjugacy classes of subgroups — too many to draw. Only its 153 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:
(2,6)(7,8)(9,10)(11,15)(12,14)(13,16)(17,18,22,19,21,24,30,26)(20,27,29,32,28,23,31,25);(1,2)(3,7)(4,8)(5,6)(9,11)(10,13)(12,16)(14,15)(17,19,22,24,21,26,30,18)(20,25,29,27,28,32,31,23);(1,3,5,4)(2,7,6,8)(10,14)(13,15)(17,20)(18,23)(19,25)(21,28)(22,29)(24,27)(26,32)(30,31);(1,4,5,3)(2,7,6,8)(11,16)(13,15)(20,28)(23,27)(25,32)(29,31)Polycyclic presentation:
2^8|a^2=g,b^2=g,c^2=ef,d^2=ef,g^2=h,b^a=be,c^a=cf,d^b=deh,d^c=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #136306, #137290, #139979, … |
| maximal normal subgroup of | #136306, #137290, #139979, … |
| maximal quotient of | #356454, #356929, #357103, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16285);