#13473

order	256
nilpotency class	4
isoclinism	isoclinic class of #13472
automorphism	Aut(G) order 4096
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#2067
commutator, Frattini, cosocle	C₂²×C₈	C₂³
prime radical	C₂³	#137
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

148 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 148 conjugacy classes of subgroups — too many to draw. Only its 53 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

34 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#13473, #499319, #499515, …
maximal normal subgroup of	C₂×#13473, #499319, #499515, …
maximal quotient of	#145905, #145914, #145925, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 6461);