#13671

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
prime radical	#3415	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #131 ◁ #504 ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 142 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:

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

Polycyclic presentation:

2^8|a^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×#13671, #512848, #512849, …
maximal normal subgroup of	C₂×#13671, #512848, #512849, …
maximal quotient of	#151179, #151181, #151291, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 6659);