Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13312 |
| automorphism | Aut(G) = #72314138 (order 4096) · Inn(G) = #469 · Out(G) = C2×C4×D4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #469 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| socle | C2 | #1999 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
260 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 260 conjugacy classes of subgroups — too many to draw. Only its 57 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 — 3 generators on 64 points:
(1,2)(3,15)(4,21)(5,26)(6,13)(7,12)(8,14)(9,17)(10,24)(11,29)(16,45)(18,51)(19,39)(20,44)(22,54)(23,56)(25,37)(27,42)(28,41)(30,40)(31,38)(32,61)(33,49)(34,50)(35,46)(36,59)(43,47)(48,64)(52,55)(53,58)(57,62)(60,63);(1,3)(2,9)(4,22)(5,27)(6,18)(7,19)(8,20)(10,35)(11,39)(12,33)(13,42)(14,34)(15,26)(16,24)(17,29)(21,52)(23,57)(25,47)(28,50)(30,49)(31,48)(32,37)(36,43)(38,62)(40,44)(41,51)(45,53)(46,59)(54,56)(55,58)(60,61)(63,64);(1,4)(2,10)(3,16)(5,23)(6,31)(7,24)(8,25)(9,32)(11,36)(12,37)(13,21)(14,38)(15,43)(17,46)(18,22)(19,47)(20,48)(26,53)(27,55)(28,60)(29,58)(30,59)(33,62)(34,52)(35,42)(39,64)(40,63)(41,56)(44,54)(45,51)(49,61)(50,57)Polycyclic presentation:
2^8|d^2=gh,e^2=h,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gh,c^b=cf,d^b=dg,f^b=fh,g^b=gh,e^c=eh,f^c=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #93768, #93769, #151185, … |
| maximal normal subgroup of | #93768, #93769, #151185, … |
| maximal quotient of | #142246, #142247, #142288, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6300);