Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33388 |
| automorphism | Aut(G) order 24576 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C22×#102 |
| commutator | C8 | C23×C4 |
| Frattini, cosocle | C2×C8 | C24 |
| prime radical | C2×#578 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×#41 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
551 conjugacy classes of subgroups; only the 221 normal subgroups are shown.
This group has 551 conjugacy classes of subgroups — too many to draw. Only its 221 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:
(1,2,6,11,7,13,27,12)(3,14,17,36,18,41,26,40)(4,9,21,34,19,32,48,35)(5,22,24,10,25,37,53,38)(8,23,29,16,31,45,30,46)(15,39,44,58,42,60,52,61)(20,49,51,33,47,59,64,57)(28,50,55,43,54,63,56,62);(1,3,7,18)(2,8,13,31)(4,15,19,42)(5,23,25,45)(6,26,27,17)(9,28,32,54)(10,36,38,40)(11,30,12,29)(14,22,41,37)(16,24,46,53)(20,50,47,63)(21,52,48,44)(33,58,57,61)(34,56,35,55)(39,49,60,59)(43,51,62,64);(1,4)(2,9)(3,15)(5,20)(6,21)(7,19)(8,28)(10,33)(11,34)(12,35)(13,32)(14,39)(16,43)(17,44)(18,42)(22,49)(23,50)(24,51)(25,47)(26,52)(27,48)(29,55)(30,56)(31,54)(36,58)(37,59)(38,57)(40,61)(41,60)(45,63)(46,62)(53,64);(4,19)(9,32)(15,42)(20,47)(21,48)(28,54)(33,57)(34,35)(39,60)(43,62)(44,52)(49,59)(50,63)(51,64)(55,56)(58,61)Polycyclic presentation:
2^8|a^2=f,b^2=h,e^2=g,f^2=h,g^2=h,b^a=be,e^a=eg,g^a=gh,e^b=egh,f^b=fh,g^b=gh,d^c=dh58 conjugacy classes — view the full character table.
| Fitting subgroup of | #33393⋊C3 |
| maximal subgroup of | #497411, #497412, #498125, … |
| maximal normal subgroup of | #497411, #497412, #498125, … |
| maximal quotient of | #481067, #481285, #481320, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26381);