#13513

order	256
nilpotency class	4
isoclinism	isoclinic class of #13392
automorphism	Aut(G) = #72312248 (order 2048) · Inn(G) = #482 · Out(G) = C₂³×C₄
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#482
commutator, Frattini, cosocle	C₂²×C₈	C₂³
socle	C₂	#2085
prime radical	C₂³	#137
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

114 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 114 conjugacy classes of subgroups — too many to draw. Only its 51 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#13513, #499418, #499595, …
maximal normal subgroup of	C₂×#13513, #499418, #499595, …
maximal quotient of	#146171, #146177, #146180, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 6501);