Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7694 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C2×C22⋊C4 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C23 ▷ C1 | |
173 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 173 conjugacy classes of subgroups — too many to draw. Only its 59 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2)(3,9,11,6)(4,12)(5,8)(7,13)(10,14,16,15)(17,18)(19,25,27,32)(20,24)(21,23)(22,26,30,28)(29,31)(33,34,36,40,37,39,45,47)(35,41,43,48,42,38,44,46);(1,3)(2,6)(4,10)(5,11)(7,14)(8,9)(12,15)(13,16)(17,19,21,27)(18,22,23,30)(20,28,29,26)(24,32,31,25)(33,35)(34,38)(36,44)(37,42)(39,41)(40,48)(43,45)(46,47);(1,4,5,13)(2,7,8,12)(3,10,11,16)(6,14,9,15)(18,23)(22,30)(24,31)(25,32)(34,39)(35,42)(40,47)(43,44)Polycyclic presentation:
2^8|a^2=e,b^2=g,c^2=fh,d^2=g,e^2=h,b^a=bd,c^a=cfg,d^a=df,c^b=ch,d^b=dg,e^b=efg46 conjugacy classes — view the full character table.
| maximal subgroup of | #97309, #97310, #97311, … |
| maximal normal subgroup of | #97309, #97310, #97311, … |
| maximal quotient of | #96296, #96756, #96758, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 717);