#13670

order	256
nilpotency class	5
isoclinism	isoclinic class of #13669
automorphism	Aut(G) = D₄×#12067 (order 2048) · Inn(G) = #2184 · Out(G) = C₂×C₈
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#2184
commutator, Frattini, cosocle	C₂×C₁₆	C₂³
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #131 ◁ #504 ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

139 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 139 conjugacy classes of subgroups — too many to draw. Only its 37 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

(2,9)(3,14)(4,18)(5,7)(6,20)(10,32)(11,37)(12,38)(13,31)(15,42)(16,46)(17,24)(19,26)(21,49)(22,48)(23,50)(25,27)(28,30)(29,33)(34,43)(35,57)(36,56)(39,53)(40,44)(45,63)(47,58)(51,55)(52,62)(59,61)(60,64);(1,2)(3,10)(4,19)(5,11)(6,26)(7,28)(8,13)(9,18)(12,20)(14,33)(15,43)(16,47)(17,36)(21,38)(22,53)(23,51)(24,35)(25,39)(27,55)(29,44)(30,48)(31,50)(32,42)(34,41)(37,49)(40,58)(45,64)(46,59)(52,61)(54,60)(56,63)(57,62);(1,3,8,17)(2,10,13,36)(4,15,23,45)(5,24,7,14)(6,16,27,46)(9,29,31,57)(11,35,28,33)(12,34,39,60)(18,44,50,62)(19,43,51,64)(20,41,25,54)(21,52,22,40)(26,47,55,59)(30,56,37,32)(38,61,53,58)(42,48,63,49)

Polycyclic presentation:

2^8|c^2=h,d^2=fg,e^2=h,f^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=df,e^a=eh,f^a=fg,g^a=gh,d^b=df,f^b=fg,g^b=gh,e^c=eh

Conjugacy classes & character table

37 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#13670, #512848, #512850, …
maximal normal subgroup of	C₂×#13670, #512848, #512850, …
maximal quotient of	#151177, #151178, #151289, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 6658);