Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7422 |
| automorphism | Aut(G) = #72285396 (order 2048) · Inn(G) = #1347 · Out(G) = C22×C4 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1347 |
| commutator | C2×C16 | C2×C4 |
| Frattini, cosocle | C2×M32 | C22 |
| prime radical | C23 | D16 |
| LCS | G ▷ C2×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C22×C8 ◁ C2×M32 ◁ G | |
| LECS | G ▷ C2×M32 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
70 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,2)(3,14,16,45)(4,20,23,44)(5,27)(6,30)(7,33)(8,13)(9,15)(10,24,39,25)(11,22)(12,29)(17,50,18,51)(19,52,34,46)(21,41)(26,59,57,53)(28,62,31,38)(32,42)(35,60)(36,54)(37,49)(40,58,63,56)(43,48)(47,64)(55,61);(1,3,8,16)(2,9,13,37)(4,21,23,47)(5,28,6,31)(7,34,29,19)(10,38,39,62)(11,41,42,64)(12,43,33,36)(14,44,45,20)(15,24,49,25)(17,22,18,32)(26,60,57,55)(27,61,30,35)(40,51,63,50)(46,53,52,59)(48,58,54,56)Polycyclic presentation:
2^8|a^2=d,b^2=h,c^2=fgh,e^2=g,f^2=g,g^2=h,b^a=bc,c^a=ce,e^a=eg,f^a=fgh,g^a=gh,c^b=cf,d^b=defgh,e^b=egh,f^b=fgh,g^b=gh,d^c=dh31 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7423, #150254, #150390, … |
| maximal normal subgroup of | C2×#7423, #150254, #150390, … |
| maximal quotient of | #93785, #93807, #93849, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 411);