#13622

order	256
nilpotency class	5
isoclinism	isoclinic class of #1418
automorphism	Aut(G) order 4096
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#357
commutator	C₁₆	C₂²×C₄
Frattini, cosocle	C₂×C₁₆	C₂³
socle	C₂	C₂×#357
prime radical	C₂×#508	C₂
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

147 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 147 conjugacy classes of subgroups — too many to draw. Only its 61 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)(3,14,17,45)(4,10)(5,21)(6,26)(7,30)(8,13)(9,16,33,51)(11,23)(12,28)(15,43,31,47)(18,54,53,64)(19,55)(20,41)(22,40,52,44)(24,60)(25,42)(27,46,50,37)(29,38)(32,48,35,62)(34,61,63,56)(36,57)(39,49)(58,59);(1,3)(2,9)(4,15)(5,22)(6,27)(7,31)(8,17)(10,32)(11,37)(12,40)(13,33)(14,42)(16,49)(18,28)(19,56)(20,53)(21,45)(23,51)(24,61)(25,52)(26,34)(29,50)(30,35)(36,54)(38,63)(39,46)(41,44)(43,60)(47,55)(48,59)(57,62)(58,64);(1,4,8,7)(2,10,13,30)(3,15,17,31)(5,19,25,24)(6,20,29,28)(9,32,33,35)(11,36,39,58)(12,26,41,38)(14,43,45,47)(16,48,51,62)(18,27,53,50)(21,55,42,60)(22,56,52,61)(23,57,49,59)(34,44,63,40)(37,54,46,64)

Polycyclic presentation:

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

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#151968, #151971, #152167, …
maximal normal subgroup of	#151968, #151971, #152167, …
maximal quotient of	#150062, #150064, #150065, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 6610);