Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33856 |
| automorphism | Aut(G) = #72265099 (order 2048) · Inn(G) = D4×D8 · Out(G) = C22×C4 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | D4×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #443 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
323 conjugacy classes of subgroups; only the 105 normal subgroups are shown.
This group has 323 conjugacy classes of subgroups — too many to draw. Only its 105 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:
(1,2,8,14)(3,15,20,50)(4,21,25,46)(5,26,31,44)(6,13,7,12)(9,34,38,18)(10,32,41,24)(11,29,45,30)(16,36,22,59)(17,35,56,39)(19,57,33,42)(23,60,61,40)(27,47,51,37)(28,43,62,64)(48,54,63,53)(49,55,58,52);(1,3)(2,9)(4,22)(5,27)(6,19)(7,33)(8,20)(10,39)(11,42)(12,37)(13,47)(14,38)(15,26)(16,25)(17,23)(18,30)(21,58)(24,54)(28,55)(29,34)(31,51)(32,53)(35,41)(36,40)(43,63)(44,50)(45,57)(46,49)(48,64)(52,62)(56,61)(59,60);(1,4)(2,10)(3,16)(5,23)(6,32)(7,24)(8,25)(9,35)(11,40)(12,46)(13,21)(14,41)(15,48)(17,51)(18,52)(19,54)(20,22)(26,43)(27,56)(28,30)(29,62)(31,61)(33,53)(34,55)(36,57)(37,58)(38,39)(42,59)(44,64)(45,60)(47,49)(50,63);(1,4)(2,10)(3,17)(5,28)(6,24)(7,32)(8,25)(9,36)(11,43)(12,21)(13,46)(14,41)(15,49)(16,51)(18,53)(19,55)(20,56)(22,27)(23,30)(26,40)(29,61)(31,62)(33,52)(34,54)(35,57)(37,63)(38,59)(39,42)(44,60)(45,64)(47,48)(50,58)Polycyclic presentation:
2^8|a^2=h,e^2=gh,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fh,g^a=gh,c^b=ch,d^b=de,e^b=eg,g^b=gh,f^c=fh,e^d=eg,g^d=gh43 conjugacy classes — view the full character table.
| maximal subgroup of | #151528, #151529, #151578, … |
| maximal normal subgroup of | #151528, #151529, #151578, … |
| maximal quotient of | #491728, #491735, #491738, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26845);