#13352

order	256
nilpotency class	4
isoclinism	isoclinic class of #13350
automorphism	Aut(G) = #72317013 (order 4096) · Inn(G) = #2021 · Out(G) = C₂²×D₄
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#2021
commutator, Frattini, cosocle	C₂×#41	C₂³
prime radical	#585	C₂²
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#41 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

341 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 341 conjugacy classes of subgroups — too many to draw. Only its 47 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:

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

Polycyclic presentation:

2^8|a^2=h,c^2=h,e^2=h,b^a=bd,c^a=ce,e^a=eh,f^a=fgh,c^b=cf,g^b=gh,d^c=dg,e^c=eh,f^d=fh

Conjugacy classes & character table

28 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#151152, #151153, #151324, …
maximal normal subgroup of	#151152, #151153, #151324, …
maximal quotient of	#142227, #142275, #142323, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 6340);