Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #576 |
| automorphism | Aut(G) = C4×#72217027 (order 6144) · Inn(G) = C22×D4 · Out(G) = C2×C4×S4 |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | C22×D4 |
| commutator | C4 | C23×C8 |
| Frattini, cosocle | C2×C8 | C24 |
| socle | C2 | C22×C22⋊C8 |
| prime radical | #585 | C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
557 conjugacy classes of subgroups; only the 261 normal subgroups are shown.
This group has 557 conjugacy classes of subgroups — too many to draw. Only its 261 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,12,7,13,27,40,8,14,28,41,5,11,25,38)(3,15,18,43,17,32,47,39,19,37,26,53,24,9,29,31)(4,10,22,35,23,36,52,61,20,33,48,57,21,34,50,59)(16,42,46,62,45,55,63,60,44,58,51,64,49,30,54,56);(1,3)(2,9)(4,16)(5,24)(6,26)(7,17)(8,19)(10,30)(11,37)(12,39)(13,15)(14,32)(18,28)(20,44)(21,49)(22,51)(23,45)(25,47)(27,29)(31,41)(33,55)(34,58)(35,60)(36,42)(38,43)(40,53)(46,48)(50,63)(52,54)(56,57)(59,62)(61,64);(1,4)(2,10)(3,16)(5,21)(6,22)(7,23)(8,20)(9,30)(11,34)(12,35)(13,36)(14,33)(15,42)(17,45)(18,46)(19,44)(24,49)(25,50)(26,51)(27,52)(28,48)(29,54)(31,56)(32,55)(37,58)(38,59)(39,60)(40,61)(41,57)(43,62)(47,63)(53,64);(4,20)(10,33)(16,44)(21,23)(22,48)(30,55)(34,36)(35,57)(42,58)(45,49)(46,51)(50,52)(54,63)(56,60)(59,61)(62,64)Polycyclic presentation:
2^8|a^2=f,e^2=h,f^2=g,g^2=h,b^a=be,e^b=eh,f^b=fh,d^c=dh88 conjugacy classes — view the full character table.
| Fitting subgroup of | #33340⋊C3 |
| maximal subgroup of | #496805, #496806, #496807, … |
| maximal normal subgroup of | #496805, #496806, #496807, … |
| maximal quotient of | #480009, #480045, #480069, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26328);