#33441

order	256
nilpotency class	3
isoclinism	isoclinic class of AGL(1,ℤ/8)
automorphism	Aut(G) = C₄×#511819 (order 2048) · Inn(G) = C₂×D₄ · Out(G) = C₄×#120
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₁₆	C₂×D₄
commutator	C₄	C₂³×C₈
Frattini, cosocle	C₂×C₈	C₂⁴
socle	C₂	C₂×C₈×D₄
prime radical	#575	C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₁₆ ◁ C₂²×C₁₆ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

323 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 323 conjugacy classes of subgroups — too many to draw. Only its 197 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 — 4 generators on 64 points:

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

Polycyclic presentation:

2^8|a^2=f,c^2=f,e^2=h,f^2=g,g^2=h,b^a=be,d^a=dh,e^a=eh,e^b=eh

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#152489, #152497, #152884, …
maximal normal subgroup of	#152489, #152497, #152884, …
maximal quotient of	#483487, #483488, #483520, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 26429);