#13544

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	#2101
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

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

normal subgroups only

This group has 106 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)(3,15,18,43)(4,10,22,37)(5,11)(6,14)(7,12)(8,13)(9,17,33,29)(16,45)(19,39,50,63)(20,52)(21,35,30,44)(23,36,55,62)(24,38)(25,42)(26,57,56,32)(27,40,46,64)(28,41)(31,49)(34,51)(47,59)(48,58)(53,61)(54,60);(1,3,8,29,6,18,5,17)(2,9,11,15,14,33,13,43)(4,20,30,51,22,54,21,53)(7,26,28,19,25,56,24,50)(10,34,35,52,37,61,44,60)(12,39,38,57,42,63,41,32)(16,46,49,55,48,27,47,23)(31,40,45,62,59,64,58,36);(1,4,5,21,6,22,8,30)(2,10,11,35,14,37,13,44)(3,16,17,47,18,48,29,49)(7,27,24,55,25,46,28,23)(9,31,15,45,33,59,43,58)(12,40,38,62,42,64,41,36)(19,51,26,20,50,53,56,54)(32,60,39,34,57,52,63,61)

Polycyclic presentation:

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

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#13544, #499560, #499595, …
maximal normal subgroup of	C₂×#13544, #499560, #499595, …
maximal quotient of	#146288, #146289, #148738, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 6532);