#12685

order	256
nilpotency class	3
isoclinism	isoclinic class of #453
automorphism	Aut(G) = C₄×#7622880 (order 2048) · Inn(G) = #120 · Out(G) = C₂⁴×C₄
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#120
commutator	C₂×C₄	C₂²×C₈
Frattini, cosocle	C₂²×C₈	C₂³
socle	C₂	#1779
prime radical	#585	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

445 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 445 conjugacy classes of subgroups — too many to draw. Only its 105 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#93679, #93680, #136829, …
maximal normal subgroup of	#93679, #93680, #136829, …
maximal quotient of	#129205, #129231, #129244, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 5673);