#13625

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

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

normal subgroups only

This group has 145 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:

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

Polycyclic presentation:

2^8|a^2=e,c^2=h,d^2=fgh,e^2=h,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, #151973, #152166, …
maximal normal subgroup of	#151968, #151973, #152166, …
maximal quotient of	#150062, #150063, #150123, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 6613);